Integration of Broadband Conductance Input in Rat Somatosensory CorticalInhibitory Interneurons: An Inhibition-Controlled Switch Between Intrinsicand Input-Driven Spiking in Fast-Spiking Cells
T. Tateno and H.P.C. RobinsonDepartment of Physiology, Development, and Neuroscience, University of Cambridge, Cambridge, United Kingdom
Submitted 21 September 2008; accepted in final form 5 December 2008
Tateno T, Robinson HPC. Integration of broadband conductanceinput in rat somatosensory cortical inhibitory interneurons: an inhibi-tion-controlled switch between intrinsic and input-driven spiking infast-spiking cells. J Neurophysiol 101: 1056–1072, 2009. First pub-lished December 17, 2008; doi:10.1152/jn.91057.2008. Quantitativeunderstanding of the dynamics of particular cell types when respond-ing to complex, natural inputs is an important prerequisite for under-standing the operation of the cortical network. Different types ofinhibitory neurons are connected by electrical synapses to nearbyneurons of the same type, enabling the formation of synchronizedassemblies of neurons with distinct dynamical behaviors. Under whatconditions is spike timing in such cells determined by their intrinsicdynamics and when is it driven by the timing of external input? In thisstudy, we have addressed this question using a systematic approach tocharacterizing the input–output relationships of three types of corticalinterneurons (fast spiking [FS], low-threshold spiking [LTS], andnonpyramidal regular-spiking [NPRS] cells) in the rat somatosensorycortex, during fluctuating conductance input designed to mimic nat-ural complex activity. We measured the shape of average conductanceinput trajectories preceding spikes and fitted a two-component linearmodel of neuronal responses, which included an autoregressive termfrom its own output, to gain insight into the input–output relationshipsof neurons. This clearly separated the contributions of stimulus anddischarge history, in a cell-type dependent manner. Unlike LTS andNPRS cells, FS cells showed a remarkable switch in dynamics, fromintrinsically driven spike timing to input-fluctuation–controlled spiketiming, with the addition of even a small amount of inhibitoryconductance. Such a switch could play a pivotal role in the functionof FS cells in organizing coherent gamma oscillations in the localcortical network. Using both pharmacological perturbations and mod-eling, we show how this property is a consequence of the particularcomplement of voltage-dependent conductances in these cells.
I N T R O D U C T I O N
The cortical network contains a variety of distinct inhibitoryneuron types, differing in electrical response properties, mor-phology, and expression of peptides and calcium-binding pro-teins (Kawaguchi 1995). Over the last ten years, it has beenshown that interneurons of a specific functional type, such asfast-spiking (FS) or low-threshold spiking (LTS) neurons, formgap junctional connections specifically with other neurons ofthe same type (Connors and Long 2004; Hestrin and Galarreta2005). Thus there are multiple electrically coupled assembliesof neurons, each with its own specific dynamical characteris-tics. This can lead to tight synchrony of firing among coupled
neurons (Mancilla et al. 2007) and suggests that the intrinsicdynamics of the neurons in each electrical network may di-rectly represent one particular motif or pattern in the repertoireof network activity. For example, we have shown that FSneurons have a hard, “type 2” threshold firing frequency at alow gamma frequency (20–30 Hz), when driven by synaptic-like conductance inputs (Tateno et al. 2004). This stronglysuggests that an electrically coupled network of FS neurons—because of the shared intrinsic integrative properties of theconnected neurons—would show stable synchronous and pe-riodic firing at this frequency, when sufficiently excited. Suchperiodic FS firing could underlie locally generated gamma-frequency oscillations (Hasenstaub et al. 2005; Morita et al.2008). Furthermore, recent studies show that, in layer 2/3barrel cortex of awake mice, electrical activity of adjacentneurons is asynchronous during an active (whisking) state,whereas the neurons show synchronous oscillations duringquiet states (Poulet and Petersen 2008). Understanding theneural mechanisms of such brain state transitions is a key tounderstanding sensory perception, sensorimotor functions, andlearning (Gilbert and Sigman 2007). It is therefore important toask: Under what conditions is spike timing in such cellsdetermined by their intrinsic dynamics and when is it domi-nated by input fluctuations? What causes an electrically cou-pled network to exert its intrinsic dynamics on the rest of thenetwork or, conversely, to be driven by activity in the network?
Herein, we have used a fluctuating conductance stimulus,with a high variability resembling that of natural synaptic input(Destexhe et al. 2001; Harsch and Robinson 2001; Softky andKoch 1993), and computed average spike-triggered conduc-tance trajectories (ASTCTs), in fast-spiking (FS), low-thresh-old spiking (LTS), and nonpyramidal regular-spiking (NPRS)neurons. The use of a conductance stimulus gives a much morerealistic spiking dynamics than does a fluctuating current input,by reproducing the shunting, saturating behavior of actualsynaptic conductance inputs and a greatly shortened, dynamicmembrane time constant, as well as spike-shape variations incertain classes of cells (de Polavieja et al. 2005). Intracellularrecordings in vivo have revealed that cortical neurons aresubjected to an intense synaptic bombardment, resulting in theresting conductance being generally much higher in the intactbrain than that in in vitro preparations (Destexhe et al. 2003).Thus neocortical networks most likely operate in a “highconductance state,” which must have profound effects on the
Address for reprint requests and other correspondence: T. Tateno, Depart-ment of Mechanical Science and Bioengineering, Graduate School of Engi-neering Science, Osaka University, Osaka, Japan, 1-3, Machikaneyama-cho,Toyonaka-shi, 560-8531 Japan.
The costs of publication of this article were defrayed in part by the paymentof page charges. The article must therefore be hereby marked “advertisement”in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
J Neurophysiol 101: 1056–1072, 2009.First published December 17, 2008; doi:10.1152/jn.91057.2008.
synaptic integrative properties of neurons. Conductance inputsare integrated quite differently within cells than are currentinputs, both actively and passively (Williams 2004), and canlead to very different membrane potential dynamics (e.g.,Fernandez and White 2008). In this study, the use of conduc-tance inputs was critical because fixed-pattern current injectioncannot adequately mimic shunting inhibitory synaptic inputs inparticular. To provide a quantitative characterization of what isresponsible for the generation and timing of spikes and howthese are influenced by the nature of the input, we fittedresponses to a two-component nonparametric linear model ofspike generation, including an autoregressive term—i.e., in-cluding its own output. Models estimated in this way fromnoisy conductance-driven activity in interneurons clearly sep-arated the contributions of stimulus and discharge history, in acell-type–dependent manner. In particular, unlike LTS andNPRS cells, FS cells showed a remarkable switch in dynamics,from intrinsically driven spike timing to input-fluctuation–controlled spike timing, with the addition of even a smallamount of inhibitory conductance. Such a transition could playa pivotal role in switching on and off the ability of FS cells toorganize coherent gamma oscillations in the local corticalnetwork. Using both pharmacological perturbation and model-ing, we show how this property is a consequence of theparticular complement of voltage-dependent conductances inthese cells.
M E T H O D S
Slice preparation and recording
All procedures involving animals were carried out in accordancewith the APS’s Guiding Principles in the Care and Use of Animalsand UK Home Office regulations and were approved by University ofCambridge and Osaka University. Transverse slices were preparedfrom somatosensory cortex of 15- to 21-day-old Wistar rats usingstandard techniques (Sakmann and Stuart 1995). During slicing, tissuewas kept in sodium-free solution that had the following composition(in mM): 254 sucrose, 2.5 KCl, 26 NaHCO2, 10 glucose, 1.25NaH2PO4, 2 CaCl2, and 1 MgCl2. Slices of 300-�m thickness werecut on a vibrating slicer (Microslicer DTK-3000, DSK, Kyoto, Japan)and kept in Ringer solution at room temperature for �2 h beforerecording. The Ringer solution contained (in mM):125 NaCl, 2.5 KCl,25 NaHCO2, 25 glucose, 1.25 NaH2PO4, 2 CaCl2, and 1 MgCl2. Bothslicing and recording solutions were equilibrated with 95% O2-5%CO2 gas to a final pH of 7.4. Slices were viewed with an uprightmicroscope (Olympus BW50WI, Olympus UK, London) using infra-red differential interference contrast optics. All experiments wereperformed at 34 � 1°C. Whole cell patch-clamp recordings weremade from the somas of neurons in layers 2 and 3, of nonpyramidalshape, and with multipolar dendrites, which were classified from theirelectrical responses (see following text) as fast-spiking, regular-spiking, or low-threshold spiking (Kawaguchi 1995). During record-ing, the slices were perfused continuously with Ringer solution inwhich 10 �M bicuculline or gabazine (Sigma), 10 �M CNQX, and 10�M AP5 (Tocris Cookson, Bristol, UK) were included to block mostsynaptic conductances. Somatic patch-pipette recordings were madewith a Multiclamp 700A amplifier (Molecular Devices, Sunnyvale,CA) in current-clamp mode, correcting for prenulled liquid junctionpotential. Whole cell recording pipettes (Clark GC150F-7.5) of 4.1- to6.3-M� resistance were filled with the standard intracellular solution(in mM): 105 K-gluconate, 30 KCl, 10 HEPES, 10 phosphocreatineNa2, 4 ATP-Mg, and 0.3 Na-GTP, balanced to pH 7.3 with NaOH.Series resistance compensation was used. Signals were filtered at 5kHz and sampled with 12/16-bit resolution at 20 kHz.
Conductance injection
For conductance injection (dynamic-clamp) stimulation (Robinson andKawai 1993; Sharp et al. 1993), an SM-1 or SM-2 conductance injectionsystem (Cambridge Conductance, Cambridge, UK) was used. The open-ing of a population of receptor channels at synapses is modeled byan excitatory (�-amino-3-hydroxy-5-methyl-4-isoxazolepropionicacid [AMPA]) receptor synaptic conductance gE(t) and an inhibitory(�-aminobutyric acid type A [GABAA]) receptor synaptic conductancegI(t). Depending on the changing membrane voltage V(t), an injectedcurrent is described by
I�t� � gE�t��V�t� � EE� � gI�t��V�t� � EI� (1)
where EE and EI are the reversal potentials for the AMPA- andGABAA-type conductances, respectively. For each cell, EI was set atthe membrane resting potential (Connors et al. 1988) and EE � 0 mV(Hollmann and Heinemann 1994; MacDermott and Dale 1987).
Current and conductance stimulus waveform
The standard injected current or conductance stimulus of broadbandnoise was 10–15 s in duration, repeated in sessions of 30 trials foreach fixed parameter set. The broadband noise component was definedby the Ornstein–Uhlenbeck process X(t) with the relationship
dX�t�
dt�
X�t�
�s
� �t� (2)
where (t) is the standard white Gaussian noise, �s is the filtering timeconstant, and is noise intensity. The broadband noise was actuallysynthesized by filtering white Gaussian noise with the followingrecursive formula
Xn �1
1 � K� 1
�s
Xn�1 � �tn� (3)
where Xn is the n th point of the noise waveform, n is the standardGaussian noise at the n th point, t is the sampling interval inmilliseconds, and K � t/�s. We varied the noise realizations fromtrial to trial by choosing different initial random number seeds. For alltrials in sessions, the filtering time constant was fixed at a timeconstant between 1 and 5 ms. For all trials of current injection stimuli,the noise intensity was 20–300 pA and for those of conductanceinjection stimuli, it was 1.0–4.0 nS. For each stimulus, the noisecomponent was superimposed on a constant-step component, whichwas 0–500 pA for current injection and 0–3.0 nS for conductioninjection. For the conductance injection stimuli, the sign of thecomposed signal should be nonnegative, so that occasional negativevalues in the input signal were truncated to zero. Similar types ofstimuli were used in Destexhe et al. (2001) and Hasenstaub et al.(2005).
Spike statistics
Spike times were measured as the times of upward zero crossing ofthe membrane potential. Instantaneous frequency (reciprocal of eachinterspike interval [ISI]) was computed from trains of action poten-tials evoked by 600-ms-duration pulses for the first, second, fourth,and last ISIs. Steady-state (SS) firing frequency was computed as theaverage of instantaneous frequency for the last three intervals of atrain. Current or conductance strength was usually progressivelyincreased or decreased in small (10- or 20-pA) steps. Initial instanta-neous frequency and steady-state firing rate were plotted as a functionof the injected current strength, to construct frequency–current (f–I)relationships. The maximum firing rate of a neuron was computedfrom the number of spikes per trial at the highest current strengthbefore depolarization block. Results are reported as means � SD.
1057INTEGRATION OF CONDUCTANCE INPUT IN INHIBITORY NEURONS
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
Membrane time constants were obtained by fitting a single-exponen-tial function to the initial part of 10 time-averaged voltage responsesto small (�20 or �10 pA), 600-ms-long hyperpolarizing currentpulses. Input resistance was calculated from Ohm’s law by dividingthe maximal average voltage deflection by the amplitude of theapplied current pulses. Action potential shape parameters were mea-sured from action potentials evoked by just-suprathreshold 200-mscurrent steps from a membrane potential near �60 mV. Currentstrength was increased in 20-pA increments to determine the thresh-old. Spike amplitude was measured as the difference between the peakand the threshold of the action potential. Spike threshold was deter-mined by finding the potential at which the second derivative of thevoltage waveform exceeded threefold its SD in the period precedingspike onset (Erisir et al. 1999). The afterhyperpolarization (AHP) wasmeasured as the difference between the spike threshold and voltageminimum following the action potential peak. Spike width was mea-sured at half the spike amplitude.
Data analysis
To characterize stochastic properties of spike trains, we first calcu-lated the mean, SD, and coefficient of variation (CV) of the ISIs, theISI histogram, and the hazard function. The ISI histogram representsan estimate of the underlying probability density function ft(t) �ft(tk�tk�1), where t � tk � tk�1 and tk and tk�1 are, respectively, thesuccessive k th and (k � 1) th spike times, under the assumption thatthe history dependence in the spike trains is Markov. The hazardfunction, which represents the instantaneous probability of a spikeoccurring in an infinitesimal time interval as a function of time sincethe previous spike, is defined as
h�t� �ft�t�
1 � �o
t
ft�u�du
(4)
It is termed “hazard” because in analysis of survival times, the hazardh(t)t may be interpreted as the risk of a failure in time interval [t, t �t), given that the system has survived up to time t (Cox and Miller1965). To calculate the hazard function, we approximate it as
Ph �ln �N0/N1�
BW(5)
where BW is the bin width in milliseconds and N0 and N1 are the sumof all subsequent bins in the simple interval histogram, with N0
including the bin whose Ph is being computed and N1 excluding it(Matthews 1996).
Average spike-triggered trajectories (ASTTs) of current or conduc-tance were calculated by averaging the broadband injected current orconductance from 100 ms before spikes to 20 ms after spikes. We alsocalculated ASTTs for subsets of spikes defined on the basis of thelength of the preceding ISI. We evaluated the statistical significance ofASTT properties by calculating 98% confidence limits for the averageand SD of the current or conductance preceding spikes, as describedby Bryant and Segundo (1976). For the average, we used the 98%band provided by �2.326 � ns/N on each side of the mean, wherens is the SD of the noise waveform and N is the number of spikes.For the SD about the average, the 98% confidence limits are given as(Dixon and Massey 1969)
ns � �1 � 2/�9N� � 2.326 � �2/�9N��3/2 (6)
Linear modeling
To examine the impact of spike history on spike generation, weestimated parameters of a linear model of spike generation, using the
spike times obtained during broadband conductance inputs. Themodel, which is the same as that used in Powers et al. (2005), is basedon the approach of Joeken et al. (1997) who extended the Wienerseries approach to neuronal system identification (Marmarelis et al.1986; Westiwick and Kearney 2003), to incorporate the effects ofprior neuronal activity. The model is described by the followingequation
Yn � � � ���1
q
a� Xn�� � ���1
p
b��Yn�� � �� � Rn (7)
where � is the expected value of Yn, q and p are positive integers thatdetermine the length of the kernels, a� and b� are unknown parametersrepresenting the estimated values of the stimulus and feedback kernelsrespectively at each time lag �, and Rn is an error term. In the model,we first defined the broadband conductance input Xn and the binaryspike output Yn. After the time axis was divided into small time binswith width t, Xn is a time series that represents the input to theneuron during the nth time interval [nt, (n � 1)t). Yn is a time seriesthat is unity if there was a spike in the nth time interval; otherwise, itis zero.
This model describes neural spike responses in a simple but reasonablyaccurate fashion, separating a component of spike generation due di-rectly to the input and a component that is due to recovery fromrefractoriness through the feedback kernel, both of which will ingeneral be varying in time. Rather than predicting an exact binaryspike train output, it determines the probability of a spike occurring ina given time bin. As described in Powers et al. (2005), this techniquecan be used to predict the spike probability or instantaneous firing rateon the basis of the stimulus kernel, the feedback kernel, or both of thekernels. Therefore we evaluated the performance of these differentmodels by calculating peristimulus time histograms describing theeffect of stimulus transients on firing probability. In addition, toreduce the number of parameters to estimate, we resampled the spiketimes and stimulus waveforms using a time interval of 0.2 ms ratherthan 0.05 ms. To estimate coefficient parameters {a�} and {b�} of themodel, we used the method described in Powers et al. (2005). Thebasic idea of the method is to minimize the sum of square error Rn
2 inEq. 7 using a least-squares approach. The calculation was performedby using MATLAB (The MathWorks, Natick, MA) on a PC. Tocharacterize feedback kernels, we used several parameters: minimumrate, zero crossing time, and slopes at the initial time and at thehalf-time of zero crossing (see Fig. 6Cb). The minimum rate isthe firing rate at which the feedback kernel gives a minimum value.The zero-crossing time is the time when the feedback kernel crosseszero for the first time. Finally, the two slopes at the initial time and thehalf-time of zero crossing represent steepness at these two time points.
R E S U L T S
Cell types in layer 2/3 of rat somatosensory cortex
On the basis of responses to injected step currents, putativeinhibitory neurons of nonpyramidal morphology with multipo-lar dendrites were recorded in layer 2 or layer 3 of somatosen-sory cortex and were classified into three groups: nonpyramidalregular-spiking (NPRS), low-threshold spiking (LTS), andfast-spiking (FS) cells (Beierlein et al. 2003; Connors andGutnick 1990; Kawaguchi 1995; Kawaguchi and Kubota 1997;Tateno and Robinson 2007), as shown in Fig. 1. This study isbased on recordings from 41 NPRS, 18 LTS, and 38 FSneurons. See Table 1 for basic firing statistics of the three celltypes. As described previously (Tateno and Robinson 2006,2007; Tateno et al. 2004), Fig. 1, A, B, and C, respectively,shows typical action potential waveforms for an NPRS cell, anLTS cell, and an FS cell at three levels of injected step current.
1058 T. TATENO AND H.P.C. ROBINSON
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
NPRS cells showed monophasic afterhyperpolarizations (AHPs) asseen in Fig. 1A, whereas LTS cells showed biphasic AHPs (Fig.1B). FS cells (Fig. 1C) showed deep AHPs, with the troughoccurring only a few milliseconds after the spike. NPRS cellsand FS cells differed in their basic electrical parameters,particularly in maximum firing rate (see Table 1). We also usedseveral other measures to distinguish NPRS and FS cells, asreported in Tateno et al. (2004). Notice that LTS cells wereeasily distinguished from the other two cell classes by low-threshold action potentials produced when stimulated fromhyperpolarizations (Beierlein et al. 2003; Fig. 2Aa in Tatenoand Robinson 2007). In addition, it is also notable that char-acteristically, RS and LTS cells have “type 1” threshold dy-namics and FS cells have “type 2” threshold dynamics (Tatenoand Robinson 2007; Tateno et al. 2004).
Average spike-triggered conductance trajectories
Figure 2A shows waveforms of evoked action potentials(Fig. 2Aa) and of an excitatory conductance stimulus (Fig.
2Ab) for an NPRS cell. Average spike-triggered conduc-tance trajectories (ASTCTs) were calculated for all cells(Fig. 2Ca), by averaging the broadband noise/synaptic con-ductance inputs over an interval of from �100 to �50 mswith respect to the time of occurrence of each spike in20 – 40 trials for each session (see METHODS). Similarly, Fig.2, Ba and Bb shows an average action potential (AAP) andthe SD about the average, respectively, computed for firingat 17.4 � 3.2 spikes/s and total 5,329 spikes, driven bybroadband excitatory conductance inputs as shown in Fig.2Ab. The AAP shows a shallow trough followed by a risingphase lasting about 2 ms, leading into the spike proper. Asshown in Fig. 2Bb, the SD about the AAP is reduced signifi-cantly, starting about 30 ms before the spike and reaches alocal minimum around the time of the AAP peak after a rapidincrease during the rising phase of spikes. The ASTCT alsoshows a shallow trough followed by a sharp peak immediatelypreceding the spike, as shown in Fig. 2Ca. Confidence limitsfor the average are shown in Fig. 2Ca (see METHODS). Figure2Cb shows that the SD about the ASTCT is reduced signifi-cantly starting about 45 ms before the spike and reaches aminimum around the time of the ASTCT peak. The confidencelimit for the SD is shown in Fig. 2Cb (see METHODS). As shownin Fig. 2Ca, the shape of the ASTCT was characterized by theduration of the trough and by its depth and the amplitude of thepeak, expressed as percentages of the baseline of the injectedconductance for firing rates at 11–20 Hz (see Tables 2, 3, and 4).In addition, the percentage depth of the prespike minimum inconductance SD was measured, as shown in Fig. 2Cb. Theproperties of ASTCTs for 12 NPRS, 8 LTS, and 15 FS cells aresummarized in Table 2.
C FS cellLTS cellBA NPRS cell
50 ms10 mV
50 ms10 mV
50 ms10 mV
FIG. 1. Firing properties of 3 classes of neurons in layer 2/3 somatosensory cortex. A: repetitive firing of a nonpyramidal regular spiking (NPRS) cell for 3different current steps of increasing amplitude (90–380 pA). B: repetitive firing of a low-threshold spiking (LTS) cell for 3 different current steps of increasingamplitude (20–200 pA). C: repetitive firing of a fast-spiking (FS) cell for 3 different current steps of increasing amplitude (70–300 pA). FS and LTS cells hadlarger afterhyperpolarizations than NPRS cells.
TABLE 1. Summary of basic statistics on NPRS, LTS, and FS cells
Input level dependence of ASTCTs: distinctive spike-generation dynamics of FS cells
Figure 3, Aa and Ab shows examples of ASTCTs and AAPs,respectively, for an NPRS cell. As the level of excitatorysynaptic input was increased (labeled “Low,” “Medium,” and“High” in Fig. 3, Aa and Ab), ASTCTs developed a deeptrough (around �20 ms in Fig. 3Aa) followed by an increas-ingly large, rapid peak immediately preceding the spike at 0 ms(dotted line), as previously reported by Tateno and Robinson(2006) for cortical RS cells. Similar characteristics of theASTCT are seen in LTS and FS cells (Tateno and Robinson2006). For the same NPRS cell, the excitatory input leveldetermines spike-shape parameters such as amplitude andwidth, as shown in Fig. 3Ab. In contrast, such spike-shapeencoding does not occur in FS and LTS cells (Tateno andRobinson 2006).
The ASTCT is partly determined by how the distance to thespike threshold varies during successive spikes for the excita-tory input, which can be characterized by interspike interval(ISI) probability density functions (normalized ISI histogramsin Fig. 3Ba) or by hazard functions (Fig. 3Bb) (see METHODS).Because of the AHP, the distance to spike threshold is largeimmediately following a spike and the hazard rate is zero or
very low, as shown in Fig. 3Bb. As the AHP decays, themembrane potential rises toward threshold, leading to a pro-gressive increase in the hazard rate (e.g., after 40 ms in Fig.3Bb). However, if the mean level of membrane depolarizationis below threshold after the AHP has completely decayed, thehazard rate levels off at a constant level and spikes then occuras a result of positive noise-induced deflections (the thin-linecurve labeled “Low” in Fig. 3Bb; see Matthews 1996; Powersand Binder 2000). For in-between input levels, the membranepotential and the hazard rate monotonically increase after aspike occurs. Thus the hazard rate function provides a quanti-tative portrait of the spike-generation dynamics in a complex,fluctuating input regime. Because, as shown in Fig. 1, theextent of AHPs in the three types of neurons is very different,the hazard rate for each cell type should reflect its specificintrinsic integrative properties.
Furthermore, the effects of inhibition are expected toshow differences among the three cell types (Tateno andRobinson 1996). Figure 3, Ca and Cb shows effects ofinhibitory synaptic conductance input on the ISI probabilitydensity and hazard functions, respectively, in an NPRS cell.Raising the level of inhibitory input reduced the peak of theISI distribution, shifting it to the right, and increased the
A(a)
(b) 200 ms10 mV
200 ms1 nS
B C
Time (ms)Time (ms)
Average
Mem
bran
evo
ltage
(mV
)
-100-100 -50-50 0 5050
-40-40
-30-30
-20-20
-10-10
0
1010
2020
Time (ms)
(a)
(b)
(a)
(b)
Vol
tage
(mV
)
-100-100 -50-50 0 5050
0
1010
2020
3030 SD
Time (ms)
Inje
cted
cond
ucta
nce
(nS
)
2 .02.0
2.52.5
3.03.0
3.53.5
4.04.0
4.54.5
5.05.0
-100-100 -50-50 0 5050
Average
Duration
Depth
PeakBaseline
Con
duct
ance
(nS
)
0
0 5.0 5.
1. 01. 0
1 5.1 5.
-100-100 -50-50 0 5050
SD
Depth
Baseline
FIG. 2. A: for an NPRS cell, an example of evoked actionpotentials (a) and the injected broadband excitatory conduc-tance input (b). B: an average action potential (a) evoked byexcitatory conductance input and the SD (b). The average tracewas computed from 31 trials. The firing rate (FR) was 13.1 �4.9 spikes/s and the total number of spikes was 4,183. Theaverage of the broadband stimuli was 3.0 nS and the SD was 1.5nS. C: average excitatory conductance changes associated withspikes (a) and the SD (b). Horizontal straight lines show thetotal mean of the whole conductance input in a, which corre-sponds to the DC component of input and the mean of the SDin b. The dotted lines in a and b show the 98% confidence limitsfor the average conductance input and the 98% confidencelimits for the SD. The 2 arrows in a indicate the duration of thetrough in the average spike-triggered conductance trajectory(ASTCT) and the arrow in b indicates the point at which the SDdrops consistently below the 98% limit.
1060 T. TATENO AND H.P.C. ROBINSON
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
length of the tail. These effects of excitatory and inhibitoryconductance inputs on both ISI probability density functions(PDFs) and hazard rates were quite similar in LTS cells(data not shown here).
In contrast, for FS cells, as a result of their intrinsic firingfrequency preference, excitatory conductance input hardly af-fects the shape of ISI PDFs and hazard rate functions as shownin Fig. 4, Aa and Ab, respectively, except in the immediateneighborhood of the peaks. However, inhibitory conductanceinput had a strong influence on the form of both functions.Even for a low level of inhibitory input (e.g., average inhibi-tory conductance of 0.5 nS and SD of 0.17 nS; labeled “Low”in Fig. 4, Ba and Bb), the peak of the ISI PDFs was muchsmaller and shifted to the right compared with the control case.As a result, in the presence of the inhibitory input, the hazardrate increases gradually and approaches a constant level for 80ms or greater intervals, as shown in Fig. 4Bb. Thus theintegrative properties of FS cells, as encapsulated by the ISIPDF/hazard rate functions, are more sensitive to inhibitoryinput than are those of NPRS and LTS cells, and more sensitive
to changes in inhibition than to changes in excitatory conduc-tance.
Spike history dependence on the ASTCTs
The prespike trough in the ASTCT (see Fig. 3Aa) mayindicate that spikes are more likely to occur in response todepolarizing fluctuations soon after a preceding hyperpolariza-tion removes some inactivation of sodium channels. However,the trough could also originate from an association between,for example, particularly long interspike intervals and a re-duced level of excitatory conductance late in the ISI. As shownin Powers et al. (2005), the latter effect can be resolved byexamining whether the form of the ASTCT differs for popu-lations of spikes grouped according to their preceding spikeinterval. Here, we divided ISIs into six groups with equalproportions as shown in Fig. 5, Aa, Ba, and Ca, for NPRS,LTS, and FS cells, respectively.
Generally, in all cell types, the ASTCTs calculated from theshorter ISIs had a larger peak amplitude and little or nopreceding trough, whereas ASTCTs calculated from the longerISIs had a smaller peak amplitude and a deeper trough. Thethree types of cells showed some clear differences. FS cellsshowed a much sharper, more compact prespike trough, ashighlighted for the 55- to 70-ms group of spikes (indicated byarrows in Fig. 5, Ab, Bb, Cb), than that of the other two celltypes. Moreover, Fig. 5, Ac, Bc, and Cc shows that dischargehistory, as reflected in the preceding ISI, influences the averageshape of action potentials in a cell-type–specific manner. InNPRS cells, membrane potential just before and after theaction potential peak varies with discharge history as shown inFig. 5Ac. In LTS cells, the membrane potential just before theaction potential (Fig. 5Bc), whereas in FS cells, the membranepotential immediately after the action potential peak (Fig. 5Cc)was most strongly influenced by discharge history.
These results indicate that in all three cell types, each rangeof ISIs is intrinsically related to a specific average conductanceinput trajectory and average evoked action potential shape. TheASTCTs reflect the contribution of all of these trajectories,weighted by the probability of different ISIs occurring. Inparticular, the AHP is likely to influence the specific shape ofthe ASTCTs associated with different ISIs. Thus ASTCTsreflect the influence of both stimulus history and dischargehistory on firing probability, in a cell-type–specific way.
TABLE 2. Summary of spike-triggered excitatory conductancetrajectories for NPRS, LTS, and FS cells
Parameter NPRS LTS FS
Number of cells 12 8 15Average firing rate, Hz 18.10 � 4.2 18.10 � 1.9 20.60 � 2.90Excitatory conductance
Values are means � SD. “Peak ratio” and “trough depth ratio” are, respec-tively, the ratios of peak amplitude and trough depth to the overall average ofconductance inputs (baseline) in percentage, as shown in Fig 2Ca. “Depthratio” is the ratio of depth to the overall SD (baseline) of conductance inputsin percentage (Fig 2Cb).
TABLE 3. Summary of feedback kernel properties for NPRS, LTS,and FS cells
Two-component linear representation of spike discharge
To understand how cells integrate broadband synaptic-likeconductance input, we need a quantitative model that is simple,but meaningful and accurate. We used a two-component linearmodel (Powers et al. 2005; also see METHODS) to model theresponse function of neurons, to obtain estimates of the effectsof stimulus and discharge history on firing probability. Asshown in Fig. 6A, the relationship between stimulus input andoutput spike probability is predicted by a stimulus kernel anda feedback kernel, which respectively correspond to movingaverage (MA) and autoregressive (AR) parameters in the ARMAmodel (Harvey 1993) (see METHODS). Each of the kernels isspecified by a series of coefficients predicting the firing probabil-
ity at particular fixed time lags (at intervals of 0.2 ms). Thestimulus kernel in the model predicts the average change infiring probability for input stimuli, whereas the feedback kernelpredicts the average change in firing probability followingprevious spikes. The coefficients of the kernels were, from aninitial set of random values, adjusted iteratively until theyminimized the error of the model prediction (Fig. 6B; seeMETHODS for details). Figure 6C shows the stimulus and feed-back kernels of the model from the discharge record of anNPRS cell for an excitatory conductance stimulus. Each pointin the kernels indicates the expected increase or decrease in thenumber of spikes during a specific short period (e.g., 0.2-msduration) compared with the average/background firing prob-
FIG. 3. In NPRS cells, analysis of evokedspike responses at 3 different levels of excita-tory conductance input and at 2 different levelsof inhibitory conductance input. Aa: averagespike-triggered excitatory conductance changesfrom �100 ms prior to the spike to 50 msafterward with a higher time resolution viewin the inset. Thick black trace: a high level ofexcitatory input (labeled “High” in plots)with average conductance �ex � 4.0 nS andthe SD ex � 1.6 nS; thick gray trace:“Medium,” �ex � 3.0 nS and the SD ex �1.2 nS; thin black trace: “Low,” �ex � 2.0nS and the SD ex � 0.6 nS. Ab: in the samecell shown in Aa, the average action poten-tials and a higher time resolution view in theinset. Ba: in the same cell shown in A,interspike interval (ISI) probability densityfunctions, which are ISI histograms normal-ized by the total ISI number. The bin size ofthe histograms was 8 ms. The FRs were, atthe high level of the conductance input,13.4 � 4.4 spikes/s; medium, 9.5 � 3.5spikes/s; low, 7.6 � 3.8 spikes/s. Bb: thehazard rates in the same cell shown in Ba.Ca: in an NPRS cell, ISI probability densityfunctions. The bin size was 8 ms. In all 3traces, the average of the excitatory conduc-tance input (�ex) was 3.6 nS and the SD (ex)was 1.2 nS. Thin black trace: in the controlcondition, the FR was 12.2 � 5.6 spikes/s.Thick black trace: a high level of the inhib-itory input (labeled “High”) with averageinhibitory conductance �in � 3.5 nS and theSD in � 1.2 nS. FR, 7.4 � 3.6 spikes/s; thinblack trace: “Low,” �in � 2.5 nS and the SDin � 0.80 nS. FR, 5.8 � 3.5 spikes/s.Cb: the hazard rates in the same cell shownin Ca.
1062 T. TATENO AND H.P.C. ROBINSON
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
ability, following a pulse of excitatory conductance stimulus.The kernels represent an impulse response function in the sensethat it represents the transient in output spike probability,which would be obtained in response to an impulse input. Asshown in Fig. 6Ca and the inset, the stimulus kernel has a sharppeak around 2–3 ms, followed by an exponential-like decay tobaseline within 10–20 ms. The first-order Wiener kernel,which corresponds to coefficients of a moving average modelwithout autoregressive terms, is indicated as the dotted trace inFig. 6Ca. Coefficients of the Wiener kernel during the first20-ms period are usually smaller than those of the stimuluskernel because the feedback kernel makes a negative contri-bution to the output firing probability of the two-componentmodel. As shown in Fig. 6Cb, for the same cell, the feedbackkernel increases monotonically with time until around 60 ms
from a negative firing rate before decaying to a baseline.Looking at the kernels in the period of the first 20 ms of Fig.6Cb, we can see that the occurrence of a single spike clearlyhas a more pronounced effect on the output firing probabilitythan does a brief unit pulse of excitatory stimulus. To charac-terize the feedback kernel, we used four parameters: the min-imum firing rate, the zero-crossing time, and slope values at theinitial time and the half-time to zero crossing (see METHODS) asshown by arrows in Fig. 6Cb.
Including the feedback kernel in the model seems to have arelatively small effect on the time course of the stimulus kernelas shown in Fig. 6Ca. Therefore it seems possible that thepredictions of the Wiener kernel model and the two-componentmodel could be similar, although this is not the case. Figure 6Dshows the difference between the Wiener kernel estimate and
FIG. 4. In FS cells, analysis of evoked spike responses at 3 different levels of excitatory conductance input and at 2 different levels of inhibitory conductanceinput. Aa: ISI probability density functions. The bin size was 8 ms. The 3 levels of the conductance input with average (�in) and the SD (ex) and the FRs are:high, �ex � 4.5 nS, ex � 1.5 nS, FR � 32.6 � 8.4 spikes/s; medium, �ex � 2.4 nS, the SD ex � 0.8 nS, FR � 16.0 � 10.1 spikes/s; low, �ex � 1.8 nS,the SD ex � 0.6 nS, FR � 9.0 � 7.9 spikes/s. b: the hazard rates in the same cell in Aa. Ba: in an FS cell, ISI probability density functions. The bin size was8 ms. In all 3 traces, the average of the excitatory conductance input (�ex) was 2.1 nS and the SD (ex) was 0.7 nS. Thin black trace: in control condition, theFR was 29.9 � 9.5 spikes/s. Thick back trace: a high level of the inhibitory input (labeled “High”) with average inhibitory conductance �in � 2.1 nS and theSD in � 0.7 nS. FR, 9.5 � 4.8 spikes/s; thin black trace: “Low,” �in � 0.5 nS and the SD in � 0.17 nS. FR, 12.1 � 5.9 spikes/s. Bb: the hazard rates inthe same cell shown in Ba.
1063INTEGRATION OF CONDUCTANCE INPUT IN INHIBITORY NEURONS
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
the stimulus kernel from the two-component model, which hasa negative-going peak lasting about 30 ms and with a similarduration to the width of the average action potential (cf., actionpotential waveform shown in the inset of Fig. 6D). Thereforeincluding spike-discharge history in the estimator is essentialfor an accurate estimation of output spike probability. Asshown in Fig. 6Ec, this scenario can be easily understood if
we compare the firing probability output of three models: the two-component model, a stimulus kernel model without the feed-back kernel, and the first-order Wiener kernel model. Thetwo-component model estimator shows rapid drops in spikerate immediately after occurrence of spikes because of theeffect of the feedback kernel (Fig. 6Ed). As a result, the outputfiring probability of the two-component model was smaller
(c)(b)(a)C
0 20 40 60 80 100 120 140 1600
100
200
300
400
500
600
700
800
900
Spi
keco
unt(
spik
es)
Interspike interval (ms)
FS cell
-100 -50 0 50 100-50
-40
-30
-20
-10
0
10
Time since previous spike (ms)
Mem
bran
evo
ltage
(mV
)
5 ms10 mV
Con
duct
anc e
(nS
)
Time since previous spike (ms)-100 -50 0 50 100
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
A
(b)
0 50 100 150 200 250 3000
50
100
150
200
250
Interspike interval (ms)
Spi
keco
unt(
spik
es)
(a)NPRS cell
-100 -50 0 50 1001.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
Con
duct
anc e
(nS
)
Time since previous spike (ms)
(c)
-100 -50 0 50 100-40
-30
-20
-10
0
10
20
Time since previous spike (ms)
5 ms10 mV
Mem
bran
evo
ltage
(mV
)
B
0 100 200 300 400 500 6000
50
100
150
200
250
300
Interspike interval (ms)
Spi
keco
unt(
spik
es)
(c)(b)(a)LTS cell
5 ms10 mV
-100 -50 0 50 100
-60
-40
-20
0
20
Time since previous spike (ms)
Mem
bran
evo
ltage
(mV
)
-100 -50 0 50 100
4.0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Time since previous spike (ms)
Con
duct
anc e
(nS
)C
ondu
ctan
c e(n
S)
FIG. 5. In NPRS, LTS, FS cells, dependence of ASTCTs and evoked action potentials on the duration of the preceding ISIs. A: NPRS cell. a: ISI histogram.The bin size was 8 ms. The FR was 13.1 � 4.9 spikes/s. b: ASTCTs. Each of the 6 colors corresponds to that in a. The average of the excitatory conductanceinput (�ex) was 3.0 nS and the SD (ex) was 1.1 nS. c: average action potentials. Each of the 6 colors corresponds to that in a. B: LTS cell. a: ISI histogram.The bin size was 8 ms. The FR was 12.3 � 7.2 spikes/s. b: ASTCTs. �ex � 1.2 nS and ex � 0.4 nS. c: average action potentials. C: FS cell. a: ISI histogram.The bin size was 8 ms. The FR was 21.2 � 12.1 spikes/s. b: ASTCTs. �ex � 3.0 nS and ex � 1.0 nS. c: average action potentials.
1064 T. TATENO AND H.P.C. ROBINSON
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
after occurrence of spikes than that of the other two models andthis led to a decreased estimation error.
Characteristics of stimulus and feedback kernels for threecell types
Figure 7A shows the stimulus and feedback kernels esti-mated from a discharge record of an NPRS cell at threedifferent levels (labeled “High,” “Medium,” and “Low”) of theexcitatory conductance input. As shown in Fig. 7Aa, thestimulus kernels all showed single peaks around 2–3 ms andfollowed by an exponential decay to baseline within 10–15 ms(cf. Fig. 6Ca). In addition, with increasing stimulus level, thepeak amplitude of the stimulus kernels increased, although thenormalized plot of the three kernels showed no significantchange in the time course (data not shown). Feedback kernelsfor the same cell are shown in Fig. 7Ab. Note that in calculatingthe feedback kernels, the average firing rate was subtracted, sothat all the feedback kernels eventually decay to the baseline
around the zero level. As shown in Fig. 7Ab, for the highexcitatory input, at the beginning the firing probability in thefeedback kernel for a first 30-ms period was very low, mono-tonically increased until around 50 ms after crossing the zerolevel at 30–40 ms, and finally decayed to the baseline. At themedium and low levels of the stimuli, the overall trend wassimilar, although the slope of the increment and the time at thepeak point differed. Thus the characteristics of both stimulusand feedback kernels depended on the mean firing rate. Inparticular, the time course of the feedback kernels clearlychanges with mean firing rate, whereas the stimulus kernelscalculated from the records of the different discharge rates arequite similar in shape.
Figure 7B shows inhibitory effects on stimulus and feedbackkernels for an NPRS cell. In the presence of the inhibitoryinput at two different levels (labeled “High inhibition” and“Low inhibition”), the stimulus and feedback kernels wereestimated from the relationship between excitatory conduc-
A
+
Stimulus kernel
Feedback kernel
Input OutputXn Yn−µ
Rn
+++ +
Error
a( )τ
b( )τ
+
B
D
(b)
C(a)
Time (ms)0 20 40 60 80 100
-20
-10
0
10
20
30
40
50
Fir
ing
rate
(spi
kes/
s/nS
)
2 ms10 spikes/s/nS
Stimulus kernel
Wiener kernel
0 10 20 30 40 500
0.51.01.52.02.53.03.54.04.55.0+10-2
Iteration number
Err
or
(Wiener kernel)-(Stimulus kernel)
0 20 40 60 80 100-10
-5
0
5
Firin
gra
te(s
pike
s/s/
nS)
Time (ms)
20 mV
20 ms
Action potential
0 ms
Spike-trigger line
E(a)
(b)
(c)
(d)
Stimuluskernel
Wiener kernel
Two-component model50 ms
20 mV
20spikes/s
50 ms
20spikes/s
50 ms
50 ms2 nS
0 20 40 60 80 100-80
-60
-40
-20
0
20
40
Fir
ing
rate
(spi
kes/
s )
Time (ms)
Feedback kernel
minimum rate
half time zero-cossing time
slope
FIG. 6. Two-component linear model. A: ablock diagram of the model. The model issimilar to an autoregressive-moving average(ARMA) model. B: error between the timeseries of the original and estimated firingprobability as a function of iteration number(see METHODS). The model was applied to adischarge record of an NPRS cell. The errorwas minimized at 10 times iteration asshown by the arrow. C: a stimulus kernel ina and a feedback kernel in b for the sameNPRS cell in B. An expanded view of thestimulus kernel is shown in the inset of a. AWiener kernel (dotted line) is also superim-posed in a. The average FR (�) was 17.2 �5.4 spikes/s. The time lag between the coef-ficients was 0.4 ms. In b, some parametersare indicated by arrows to characterize feed-back kernels. D: difference between the Wie-ner kernel and the stimulus kernel in Ca. Atypical waveform of action potentials isshown in the inset. E: conductance stimulusin a, membrane potential in b, firing-proba-bility output of the 2-component model (thinline), the stimulus kernel model in the ab-sence of the feedback kernel (thick line), afirst-order Wiener-kernel model (dotted line)in c, and firing-probability output of thefeedback kernel model in the absence ofthe stimulus kernel in d. Triangles show thetiming of recorded action potentials. Thestimulus kernel model sometimes overesti-mated the firing probability immediately af-ter a spike, whereas the 2-component modeldid not, as shown in Ec.
1065INTEGRATION OF CONDUCTANCE INPUT IN INHIBITORY NEURONS
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
tance input and its spike discharge. As the level of the inhib-itory input increased, the inhibitory input decreased the peakamplitude of the stimulus kernels, as shown in Fig. 7Ba. Incontrast, Fig. 7Bb shows that the inhibitory input relativelyincreased the firing probability of the feedback kernels 50ms. This result indicates that recovery from spike refractori-ness in the feedback kernels was accelerated by the inhibitoryinput, enabling NPRS cells to fire sooner after the previousspike.
Figure 8A shows the stimulus and feedback kernels esti-mated from the discharge record of an FS cell at three differentlevels (labeled “High,” “Medium,” and “Low”) of the excita-tory conductance input. Stimulus kernels (Fig. 8Aa) weresimilar to those of NPRS cells. Feedback kernels for the samecell are shown in Fig. 8Ab. All feedback kernels eventuallydecayed to the baseline around the zero level. However, thefirst 30-ms period of each kernel is different. For the highexcitatory input, the firing probability in the feedback kernel isvery low in the first 8 ms, then rapidly increases to cross zeroat around 15 ms, before finally decaying to the baseline. At themedium level of the stimulus, the overall trend was similar,although the rising slope was reduced and the time of the peakwas shifted later.
Figure 8B shows inhibitory effects on stimulus and feedbackkernels for an FS cell. In control and in the presence of theinhibitory input with two different levels of inhibition (labeled“High inhibition” and “Low inhibition”), the stimulus andfeedback kernels were estimated from the relationship betweenexcitatory conductance input and its discharge history. As thelevel of inhibition increased, the peak of the stimulus kernelsdecreased in amplitude and shifted leftward in time, as shown
in the inset of Fig. 8Ba. Figure 8Bb shows the effect ofinhibition on the feedback kernels. Administering inhibitoryinput to FS cells drastically changed the time course of thefeedback kernels, even though the level of the input wasrelatively small (in Fig. 8B, “Low inhibition” level inputstatistics: average �in � 1.8 nS and SD in � 0.6 nS).Inhibitory input to FS cells thus has a much more powerfuleffect on the feedback kernel, accelerating the recovery fromrefractoriness, than that in NPRS cells.
The results obtained from LTS cells are similar to those fromNPRS cells, as shown in Fig. 9. However, all the feedbackkernels in LTS cells increased more slowly to the zero levelthan those of NPRS cells over the first 60 ms, as shown in Fig.9Ab. The effect of inhibition on the feedback kernels wasto shift them upward as the inhibitory input level increased(Fig. 9Bb).
Effects of changes in the AHP on the stimulusand feedback kernels
The decrease in firing probability measured by the feedbackkernel is likely to represent the influence of the postspike AHP,especially in FS cells. Although, in response to the excitatoryconductance injection, the mean discharge rate was nearlyconstant at 13–15 spike/s, changes in AHP should profoundlyinfluence the feedback kernel. In addition, the precise spiketiming evoked by the conductance input signal may be influ-enced by the change in AHP. It is known that a low tetraeth-ylammonium (TEA) concentration ( 1 mM) in the extracel-lular solution blocks a small fraction of K� channels andimpairs action potential repolarization (Erisir et al. 1999). We
A(a) (b)
B (b)
NPRS cells
0 20 40 60 80 100-20
-10
0
10
20
30
40
50
60
70
Time (ms)
Fir
ing
rate
(spi
kes/
s/nS
)
High
Medium
Low
2 ms10 spikes/s/nS
0 20Time (ms)
0 20 40 60 80 100-120
-100
-80
-60
-40
-20
0
20
40
Fir
ing
rate
(spi
kes/
s)Time (ms)
MediumLow
High
2 ms10 spikes/s
HighMediumLow
0 20Time (ms)
(a)
Fir
ing
rate
(spi
kes/
s/nS
)
Time (ms)0 20 40 60 80 100
-10
0
10
20
30
40
2 ms5 spikes/s/nS
Control
High inhibition
Low inhibition
0 20Time (ms)
Fir
ing
rate
(spi
kes/
s )
Time (ms)0 20 40 60 80 100
-50
-40
-30
-20
-10
0
10
High inhibition
Control 2 ms5 spikes/s
High inhibition
Control
Low inhibition
Low inhibition
0 20Time (ms)
FIG. 7. Stimulus and feedback kernelsfor NPRS cells. The time lag between coef-ficients of the kernels was 0.4 ms. A: in thesame NPRS cell as in Fig. 3B, the stimuluskernels in a and feedback kernels in b areshown for 3 different levels of the excitatoryconductance input. Thin line trace: “High”level of the input with average conductance�ex � 4.0 nS and the SD ex � 1.6 nS. Thickline trace: “Medium,” �ex � 3.0 nS and theSD ex � 1.2 nS. Dotted line trace: “Low,”�ex � 2.0 nS and the SD ex � 0.6 nS. TheFRs were, at the high level of the conduc-tance input, 13.4 � 4.4 spikes/s; medium,9.5 � 3.5 spikes/s; low, 7.6 � 3.8 spikes/s.B: for the same NPRS cell in Fig. 3C, thestimulus kernels in a and feedback kernels inb are shown in control and with 2 differentlevels of the inhibitory conductance input. Inthe 3 cases, the average of the excitatoryconductance input (�ex) was the same and3.6 nS and the SD (ex) was 1.2 nS. Thickline trace: in control condition, the FR was12.2 � 5.6 spikes/s. Dotted line trace: a highlevel of the inhibitory input (labeled “Highinhibition”) with average inhibitory conduc-tance �in � 3.5 nS and the SD in � 1.2 nS.FR, 7.4 � 3.6 spikes/s; thin line trace: “Lowinhibition,” �in � 2.5 nS and the SD in �0.80 nS. FR, 5.8 � 3.5 spikes/s.
1066 T. TATENO AND H.P.C. ROBINSON
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
tested the effect of perturbing the AHP in this way on thefeedback kernel and on spike timing by adding 1 mM TEA tothe extracellular solution during conductance stimuli (Fig. 10C).Erisir et al. (1999) reported that this concentration of TEAreduces the extent of the AHP by producing a partial bock of
several known K� channels, including Kv3.1–Kv3.2, leads toan increase in spike width in response to a 200-ms currentpulse and slightly decreases the instantaneous firing rate of thefirst several spikes in response to a 600-ms current pulse (seealso Table 5). First we measured the effect of TEA on the firing
(a) ( b)A
0 20 40 60 80 100
5 spikes/s/nS
2 ms
Fir
ing
rate
(spi
kes/
s/nS
)
Time (ms)
-40
-20
0
20
40
60
80 High
Medium
Low
B (a) (b)
FS cells
0 20Time (ms)
0 20 40 60 80 100-20
0
20
40
60
80
100
20 spikes/s/nS2 ms
Control
Low inhibition
High inhibition
Fir
ing
rate
(spi
kes/
s/nS
)
Time (ms)
0 20Time (ms)
Fir
ing
rate
(sp
ike
s/s )
Time (ms)0 20 40 60 80 100
-70-60-50-40-30-20-10
01020
High
Medium
Low
2 ms5 spikes/s
High
Medium
Low
0 20Time (ms)
0 20 40 60 80 100-100
-80
-60
-40
-20
0
20
40 Control Low inhibition
Highinhibition
Fir
ing
rate
(sp
ike
s/s )
Time (ms)
10 spikes/s2 ms
ControlLow inhibitionHigh inhibition
0 2 0Time (ms)
FIG. 8. For FS cells, stimulus and feed-back kernels. The time lag between the co-efficients of the kernels was 0.4 ms. A: in thesame FS cell in Fig. 4A, the stimulus kernelsin a and feedback kernels in b are shown forthe 3 different levels of the excitatory con-ductance input. Dotted line trace: “High”level of the input with average conductance�ex � 4.5 nS and the SD ex � 1.5 nS. Thinline trace: “Medium,” �ex � 2.4 nS and theSD ex � 0.8 nS. Thick line trace: “Low,”�ex � 1.8 nS and the SD ex � 0.6 nS. TheFRs were, at the high level of the conduc-tance input, 32.6 � 8.4 spikes/s; medium,16.0 � 10.1 spikes/s; low, 9.0 � 7.9spikes/s. B: for the same FS cell in Fig. 4B,the stimulus kernels in a and feedback ker-nels in b are shown in control and 2 differentlevels of the inhibitory conductance input. Inthe 3 cases, the average of the excitatoryconductance input (�ex) was the same and2.1 nS and the SD (ex) was 0.7 nS. Thickline trace: in control condition, the FR was29.9 � 9.5 spikes/s. Dotted line trace: a highlevel of the inhibitory input (labeled “Highinhibition”) with average inhibitory conduc-tance �in � 2.1 nS and the SD in � 0.7 nS.FR, 9.5 � 4.8 spikes/s. Thin line trace: “Lowinhibition,” �in � 0.5 nS and the SD in �0.17 nS. FR, 12.1 � 5.9 spikes/s.
(a)A (b)
High
Fir
ing
rate
(sp
ike
s/s)
Time (ms)0 20 40 60 80 100
-50
-40
-30
-20
-10
0
10
Low
Medium
(b)B (a)
LTS cells
Fir
ing
rate
(spi
kes/
s/nS
)
Time (ms)0 20 40 60 80 100
-20
0
20
40
60
80 High
MediumLow
2 ms
10 spikes/s/nS
0 20Time (ms)
Fir
ing
rate
(spi
kes/
s/nS
)
Time (ms)0 20 40 60 80 100
-10
0
10
20
30
40
50 Control
Low intensity
High inhibition
2 ms
4 spikes/s/nS
0 20Time (ms)
Fir
ing
rate
(sp
ike
s/s)
Time (ms)0 20 40 60 80 100
-60-50-40-30-20-10
01020
Lowinhibition
High inhibition
Control
Low inhibition
Control
Highinhibition
2 ms2 spikes/s
0 20Time (ms)
FIG. 9. Stimulus and feedback kernelsfor LTS cells. The time lags between coef-ficients of the kernels was 0.4 ms. A: stim-ulus kernels in a and feedback kernels in bare shown for the 3 different levels of theexcitatory conductance input. Thick linetrace: “High” level of the input with averageconductance �ex � 1.5 nS and the SD ex �0.5 nS. Thin line trace: “Medium,” �ex �1.0 nS and the SD ex � 0.35 nS. Dottedline trace: “Low,” �ex � 0.8 nS and the SDex � 0.25 nS. The FRs were, at the highlevel of the conductance input, 14.1 � 0.8spikes/s; medium, 10.2 � 0.8 spikes/s; low,6.4 � 0.9 spikes/s. B: for an LTS cell, thestimulus kernels in a and feedback kernelsin b are shown for control and 2 differentlevels of the inhibitory conductance input.In all 3 cases, the average of the excitatoryconductance input (�ex) was the same andthe mean was 2.4 nS and the SD (ex) was0.6 nS. Dotted line trace: in control condi-tion, the FR was 14.9 � 2.6 spikes/s. Thinline trace: a high level of the inhibitoryinput (labeled “High inhibitory”) with aver-age inhibitory conductance �in � 2.4 nS andthe SD in � 0.6 nS. FR, 9.7 � 2.4 spikes/s.Thick line trace: “Low inhibition,” �in �1.2 nS and the SD in � 0.3 nS. FR, 11.1 �2.5 spikes/s.
1067INTEGRATION OF CONDUCTANCE INPUT IN INHIBITORY NEURONS
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
rate, AHP, spike width, and other spike-shape properties. Asshown in Fig. 10A, the firing rate in FS cells was decreasedafter adding 1 mM TEA to the extracellular solution in all cases(n � 10, see Fig. 10B). Spike width was increased in all casesand this led to a reduction in amplitude (i.e., depolarization) ofthe AHP (Table 5). In Fig. 10B, a typical example of the effectof TEA on the average first action potential (20 traces) duringa 200-ms current step injection in FS cells is shown before(“Control”), during (“TEA”), and after TEA application
(“Washout”). Other spike shape properties are summarized inTable 5.
Figure 10, Ca and Cb shows, respectively, an example of anexcitatory conductance input waveform and the membranepotential trajectory (with curtailed spikes) in response to theinput before and during application of 1 mM TEA. Althoughthe conductance input trajectory was exactly the same (Fig.10Ca), AHPs are reduced during the TEA application. Thisreduction of AHPs is seen to be associated with a considerable
B
A
20 mV50 ms
Washout
1mM TEA
Control(a)
(b)
(c)
1 ms
10 mV
Washout
1mM TEA
Control
D
E
F
0 20 40 60 80 100 1200
10
20
30
40
50
60
70
80
1mM TEA
Washout
Control
Time since previous spike (ms)
Haz
ard
rate
(spi
kes/
s)
Interspike interval (ms)
Per
cent
age
ofin
terv
al(%
)
0 20 40 60 80 100 120 140150
0
5
10
15
20
Washout
1mM TEA
Control
0 10 20 30 40 50 60 70 80 90 100-10
-5
0
5
10
15
20
25
30
35
4 ms
5 spikes/s/nS
Fir
ing
rate
(spi
kes/
s)
1mM TEA
Control
Washout
(a)
0 10 20 30 40 50 60 70 80 90 100-120
-100
-80
-60
-40
-20
0
20
40
60
80
4 ms
10 spikes/s/nS
Time (ms)(b)
Time (ms)
Fir
ing
rate
(spi
kes/
s)
Washout
1mM TEA
Control
C (a)
(b)1 nS
100 ms
100 ms50 mV
1mM TEA
Control
FS cells
FIG. 10. Influence of 1 mM tetraethylammonium (TEA) on firing in FS cells. A: repetitive firing of an FS cell in control (a), in 1 mM TEA application (b),and in washout conditions (c) for a current step of amplitude 200 pA. B: 3 membrane-potential averages of repetitive firing for 150-pA current steps in control(thin line), TEA (thick line), and washout (dotted line). C: driven by excitatory conductance input (a), 2 membrane potential waveforms in the range of �40to 0 mV were shown in control (b, dotted line) and in TEA application (b, thin line). D: ISI histograms in control (thin line), TEA application (thick line), andwashout (dotted line). E: hazard functions in control (thin line), TEA application (thick line), and washout (dotted line). The data analyzed were the same asshown in E. F: stimulus kernels in control (thin line), TEA application (thick line), and washout (dotted line). G: feedback kernels in control (thin line), TEAapplication (thick line), and washout (dotted line).
TABLE 5. Summary of spike properties for 10 FS cells in control, 1 mM TEA application, and washout conditions
change in spike timing, at intermediate levels of the conduc-tance input signal such as this.
Figure 10, D and E shows the effect of TEA on dischargestatistics. The ISI histograms before (“Control”) and after TEA(“Washout”) were distributed more widely than those obtainedduring TEA application (“TEA”) although the average ISI isnot significantly different: 89.8 � 35.4 ms in control, 91.9 �35.2 ms in washout condition versus 94.5 � 35.1 ms in TEAapplication (P 0.01). The effects of TEA on the ISI histo-gram are reflected in a more rightward rising hazard functionthan those in the control and washout conditions, as shown inFig. 10E.
Figure 10, Fa and Fb shows, respectively, the stimuluskernels and feedback kernels estimated from firing evoked bythe broadband conductance input (e.g., Fig. 10Ca) before,during, and after TEA application. The stimulus kernels are notsignificantly changed in the three cases, but the peak during 1mM TEA application is shifted slightly rightward ( 1.5 ms).However, the firing probability of the feedback kernel in TEAapplication decreased more than that in the control and wash-out conditions. The firing probability in TEA remained de-pressed well below the control case, for some 50 ms, beforereturning to the same zero level. In FS cells, this effect impliesthat excitability in TEA was lower than that in the control andwashout conditions. As a result, spike history has a largerinfluence on the occurrence of the next spike during TEAapplication than in the control and washout conditions.
D I S C U S S I O N
Recent studies have shown that inhibitory interneurons inthe somatosensory cortex have a variety of molecular, electro-physiological, and morphological characteristics, which couldunderlie distinct roles in the cortical networks (Kawaguchi andKubota 1997; Markram et al. 2004; Toledo-Rodriguez et al.2005). In this study, we have examined membrane excitabilityproperties and input–output functions, to gain insight into theinfluence of naturalistic conductance-input signals and prioractivation history on spike generation in three different types ofinterneurons: NPRS, FS, and LTS cells.
Analysis of neural discharges driven by broadband or white-noise signals has been widely used to investigate the responsefunctions of neurons or the features of input that are relevant inthe spike-triggering process (Bryant and Segundo 1976;Mainen and Sejnowski 1995; Powers et al. 2005; Tateno andRobinson 2006). In the present study, with the aim of using amore naturalistic stimulus mimicking in vivo–like synapticinput, we applied fluctuating excitatory and inhibitory conduc-tance input rather than current input. We used independentOrnstein–Uhlenbeck processes to determine excitation andinhibition, to create a wide and complex range of conductancetrajectories with in vivo–like statistics of excitation and inhi-bition, but without introducing specific assumptions aboutcorrelations between them, or oscillatory dynamics, in theinput. We are interested how, in general, the intrinsic proper-ties of individual cells can organize coherent output firing fromcomplex, stochastic input. It has been shown that duringactivity in vivo, synaptic input to cortical neurons may con-tribute 80% of the total conductance of the membrane andthat injecting stochastic excitatory and inhibitory conductancesreproduces in vivo–like membrane potential distributions (Des-
texhe et al. 2001). The synaptic conductance input itself radi-cally alters the basic electrical parameters of the membrane,such as time constant, membrane resistance, and spatial atten-uation. In studying synaptic integration and spike generation itis thus extremely important to stimulate with conductancepatterns rather than current patterns, to force the neuron’sspike-generation mechanisms into a much more in vivo–likeelectrical state. This study goes beyond previous work, first, bysystematically comparing the integrative properties of differentclasses of inhibitory interneuron when driven by electricallyrealistic fluctuating conductance and by applying a two-com-ponent autoregressive model previously applied to spikinggenerated by current fluctuations to describe conductance-driven spiking, allowing the contributions of input conductanceand spike discharge history to be separated. The use of con-ductance injection here, although technically more demandingthan current injection, was crucial because it is impossible tosimulate a controlled increase in shunting inhibitory conduc-tance using a predetermined pattern of current injection.
Dependence of ASTCT properties on cell type and onspike history
ASTCTs describe the mean conductance input precedingspike generation and allow the possibility of studying howmembrane properties shape integration of naturalistic conduc-tance inputs into action potentials (Harsch and Robinson2000). Overall, ASTCTs for excitatory, AMPA receptor-likeconductance in these cell-types were characterized by a shal-low trough followed by a sharp peak just prior to spike onset(Fig. 2Ca) as we reported previously (Tateno and Robinson2006). In addition, the results of this study demonstrated thatthe time course of the ASTCTs reflects a combination of theeffects of the conductance stimulus input and recent dischargehistory on firing probability.
We computed ASTCTs for subsets of interspike intervals ofdifferent durations and found that a deep trough (1–2 nS) waspresent only for short preceding ISIs in NPRS and LTS cells,but for all durations of ISIs in FS cells. A previous study usingfluctuating current stimulation (Powers et al. 2005) found thata prespike dip in excitatory current was associated only withlonger ISIs, in regularly spiking pyramidal neurons. Thisdiscrepancy could reflect a fundamental difference in the dy-namics of these cells from NPRS and LTS cells, as well as thedifference between broadband current and conductance injec-tion. It is also reported that the hyperpolarizing trough inaverage current trajectories reflects the removal of sodiuminactivation (Fellous et al. 2003; Gutkin et al. 2003; Mainenand Sejnowski 1995) and/or calcium-activated potassium cur-rent underlying the postspike AHP (Powers et al. 2005). For FScells as well as LTS cells, in particular, the effects of dischargehistory are likely to be largely mediated by the specific types(Kv3.1–3.2) of voltage-dependent potassium channels under-lying the postspike AHP. This is less likely to be the case forNPRS cells because they have shallower AHPs than those inFS and LTS cells.
The ASTCT calculated from all spikes at a given meandischarge rate reflects the combination of the conductancetrajectories associated with particular ISIs, weighted by theprobability of occurrence of particular intervals (Aguera yArcas and Fairhall 2003; Pillow and Simoncelli 2003). The ISI
1069INTEGRATION OF CONDUCTANCE INPUT IN INHIBITORY NEURONS
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
distribution itself reflects the influence of the AHP on dis-charge probability (Matthews 1996; Powers and Binder 2000).Thus the ASTCT and the related first-order Wiener kernelreflect the influence of both stimulus and discharge history onfiring probability in interneurons. For this reason, a two-component model is useful for dissecting the contribution ofthe two influences separately.
Influence of stimulus and discharge history in thetwo-component model
In general, one method for specifying the input– outputrelationship of a system to predict the response of thesystem to arbitrary inputs is to determine the system func-tion from the output response, by using appropriate excita-tion input to the system. A nonparametric or “black box”approach to the system identification problem can be used todetermine the system transfer characteristic without specifyingthe internal structure or mechanisms (Turker and Powers2005). In particular, white noise or Wiener kernel analysis is awell-known nonparametric approach to such system identifi-cation (French and Marmarelis 1995; Marmarelis and Mar-marelis 1978; Westwick and Kearney 1998) that has been usedin the study of neurophysiological systems such as visual(Marmarelis and Naka 1972; Sakai 1992), auditory (Egger-mont 1993), and mechanoreceptor (Dickinson 1990; Frenchand Wong 1977; French et al. 2001; Kondoh et al. 1995) motorsystems (Gamble and DiCaprio 2003; Powers et al. 2005). Theactual mechanisms are replaced with a linear/nonlinear filter ora series of kernels with exactly the same transfer characteristicsas those of the system under study. In addition, it is pointed outthat such filters are closely related to the ASTCTs of neuro-physiological systems (Powers et al. 2005). However, theinput–output relationship taking account for only the inputstimulus is not enough to fully describe the system whoseinternal states depend on the previous states themselves. There-fore using a two-component model previously applied to fluc-tuating current-driven spiking (Joeken et al. 1997; Powers et al.2005; Truccolo et al. 2005), we separated the effects ofconductance stimulus and discharge history in the spike trains.In this model, the effects of discharge history are mainlyrepresented by a feedback kernel, whereas the effects of stim-ulus history are principally represented by a stimulus kernel.
The stimulus kernels predicted a brief ( 8 ms) and rapidincrease in firing probability following brief fluctuations in theconductance stimulus. In the absence of the feedback kernel inthe model, the stimulus kernel (or standard first-order Wienerkernel) showed a greater decrease in firing probability after thepeak than did the stimulus kernels of the two-component modeland thus the Wiener kernel encapsulates both stimulus anddischarge history. When shunting inhibition is added, the peakamplitude of the stimulus kernels decreased as the intensity ofinhibitory input increased, for all the cell types. In this sense,the effect of the inhibitory input on the stimulus kernels issimply to counteract proportionately the effect of the excitatoryinput.
In contrast to the stimulus kernels, the properties of feedbackkernels depended strongly on the cell type. For excitatory inputin NPRS and LTS cells, the feedback kernels predicted aprolonged ( 70-ms) decrease in firing probability followinga spike, whereas, for FS cells, the feedback kernels showed a
relatively short lasting ( 40-ms) negative phase. In all threecell types, higher levels of excitatory input caused a shorteningof the negative phase, often followed by a positive overshoot(Figs. 7Ab and 8Ab), the overshoot time depended on the celltype. Moreover, the effect of simultaneous inhibition wasdifferent, eradicating the positive overshoot in FS cells (Fig.8Bb), whereas for NPRS and LTS cells, causing a simpleupward shift of the kernel with increasing inhibition (Figs. 7Bband 9Bb). Diminishing the AHP current in FS cells by adding1 mM TEA to the bath solution led to a profound decrease inthe amplitude of the feedback kernel, but had relatively minoreffects on the stimulus kernel, consequently changing the spiketiming driven by the same conductance stimuli.
Powers et al. (2005) concluded, on the basis of the defi-ciency of prediction simply by the first-order Wiener kernel,that the linear prediction of firing probability from fluctuatingcurrent input should rather be taken as the sum of the predictedchanges based on both stimulus and feedback kernels. Theyalso stated that prediction using the first-order Wiener kernelalone underestimated the increase in firing probability pro-duced by a depolarizing input and overestimated the decreaseproduced by a hyperpolarizing input. In this study, although weused a more naturalistic conductance input, the situation wasquite similar. The two-component model yielded more accu-rate spike timing than did the stimulus kernel alone or thefirst-order Wiener kernel. We also tested another model con-sisting of both first-order and second-order Wiener kernels,without a feedback kernel, to the data. However, the results ofthis second-order Wiener-kernel model were worse than thoseof the linear two-component model (data not shown here). Thisis a further indication that the discharge history is necessary topredict precise spike timing in cortical interneurons.
An inhibition-controlled switch between intrinsically timedand input-timed spike firing in FS cells:functional implications
FS cells have several distinctive electrical features thatdistinguish them from other neurons and other inhibitory in-terneurons in particular: the ability to fire rapidly with littlefatigue, a deep AHP, and narrow spike—properties that inhippocampal FS cells appear to be conferred by a large densityof Kv 3.1/3.2 channels (Lien and Jonas 2004). It is thusreasonable to suppose that these adaptations subserve a partic-ular function in the network. There are several clues that pointto a leading role for FS neurons in the generation of gammaoscillations. Gamma oscillations, in the frequency range 30 to80 Hz, are a prominent form of synchronization in the awakecortex and are widely believed to underlie various neurocog-nitive functions, including feature binding, selective attention,and consciousness (Kastner and Ungerleider 2000; Singer andGray 1995; Tiitinen et al. 1993). Gamma rhythms most oftenoccur in bursts lasting 100–200 ms and can be generatedlocally in the cortical network, even in small pieces of in vitrocortical tissue (Cunningham et al. 2003; Shu et al. 2003). FSneurons have a hard, “type 2” threshold (Tateno et al. 2004) forexcitatory conductance input, firing periodically at a gammafrequency at the onset of spiking. Intriguingly, this critical-threshold frequency can be modulated within the gamma rangeby the level of simultaneous shunting inhibitory input (Tatenoet al. 2004). Furthermore, FS neurons are interlinked by a
1070 T. TATENO AND H.P.C. ROBINSON
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
specific, gap-junctional network that promotes a high degree ofsynchrony in their spike timing (Galarreta and Hestrin 2002;Gibson et al. 2005). The predominant gamma oscillatory com-ponent of synaptic input into the principal pyramidal cellsappears to be inhibition arising from the FS cell population(Hasenstaub et al. 2005; Morita et al. 2008). Spike timing ofprincipal cells in awake and active states appears to be stronglydetermined by the dynamics of inhibitory synaptic conductance(Rudolph et al. 2007). Recent studies show that specific infor-mation-rich signals generate relatively high frequency, includ-ing gamma range, asynchronous synaptic inputs to corticalneurons that dominate during active brain states, whereasneurons show rather synchronous membrane voltage activity inquiet states (Petersen 2007; Poulet and Petersen 2008). Under-standing the neural mechanism of such brain state transitions iscrucial for understanding neurocognitive functions.
In this study, we have demonstrated that shunting inhibitoryconductance input to FS neurons converts their integrativefunction from one in which timing of spikes is determined bythe recent spiking history, i.e., an intrinsically timed rhythmcorresponding to the steep control feedback kernel in Fig. 8Bb,to one in which it is driven directly by conductance inputfluctuations (the dramatic flattening of the feedback kernel byinhibition in Fig. 8Bb). In the context of the putative role of FSneurons in organizing gamma rhythms, this inhibition-con-trolled switch could be responsible for initiation and termina-tion of local gamma bursts. With little or no inhibitory input toFS neurons, their firing organizes into locally synchronousgamma rhythms. With even a small asynchronous inhibitoryinput, though, the integrative dynamics demonstrated hereshould cause the firing of individual FS neurons to becomemore driven by excitatory input fluctuations, dispersing thesynchrony, and thereby terminating the gamma burst.
A C K N O W L E D G M E N T S
Present addresses of T. Tateno: Department of Mechanical Science andBioengineering, Graduate School of Engineering Science, Osaka University,Osaka, Japan 1-3, Machikaneyama-cho, Toyonaka-shi, 560-8531 Japan andPRESTO, Japan and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama,Japan.
G R A N T S
This work was supported in part by a Japan Science and TechnologyAgency Precursory Research for Embryonic Science and Technology grant, bya Japanese Ministry of Education, Science, Sports and Culture Grant-in-Aid,by an Okawa Information and Telecommunication grant, and by EuropeanCommission Framework Programme 6 (FP6) research grant (NEURO project).
R E F E R E N C E S
Aguera y Arcas B, Fairhall AL. What causes a neuron to spike? NeuralComput 15: 1789–1807, 2003.
Azouz R, Gray CM. Cellular mechanisms contributing to response variabilityof cortical neurons in vivo. J Neurosci 19: 2209–2223, 1999.
Azouz R, Gray CM. Dynamic spike threshold reveals a mechanism forsynaptic coincidence detection in cortical neurons in vivo. Proc Natl AcadSci USA 97: 8110–8115, 2000.
Beierlein M, Gibson JR, Connors BW. Two dynamically distinct inhibitorynetworks in layer 4 of the neocortex. J Neurophysiol 90: 2987–3000, 2003.
Boer R, Kuyper P. Triggered correlation. IEEE Trans Biomed Eng. 15:169–179, 1968.
Bryant HL, Segundo JP. Spike initiation by transmembrane current: awhite-noise analysis. J Physiol 260: 279–314, 1976.
Connors BW, Gutnick MJ. Intrinsic firing patterns of diverse neocorticalneurons. Trends Neurosci 13: 99–104, 1990.
Connors BW, Gutnick MJ, Prince DA. Electrophysiological properties ofneocortical neurons in vitro. J Neurophysiol 48: 1302–1320, 1982.
Connors BW, Long MA. Electrical synapses in the mammalian brain. AnnuRev Neurosci 27: 393–418, 2004.
Connors BW, Malenka RC, Silva LR. Two inhibitory postsynaptic poten-tials, and GABAA and GABAB receptor-mediated responses in neocortex ofrat and cat. J Physiol 406: 443–468, 1988.
Cox DR, Miller HD. The Theory of Stochastic Processes. London: Chapman& Hall, 1965.
Cunningham MO, Davies CH, Buhl EH, Kopell N, Whittington MA.Gamma oscillations induced by kainate receptor activation in the entorhinalcortex in vitro. J Neurosci 23: 9761–9769, 2003.
Destexhe A, Rudolph M, Fellous JM, Sejnowski TJ. Fluctuating synapticconductances recreate in vivo-like activity in neocortical neurons. Neuro-science 107: 13–24, 2001.
Dickinson MH. Linear and nonlinear encoding properties of an identifiedmechanoreceptor on the fly wing measured with mechanical noise stimuli. JExp Biol 151: 219–244, 1990.
Eggermont JJ. Wiener and Volterra analyses applied to the auditory system.Hear Res 66: 177–201, 1993.
Erisir A, Lau D, Rudy B, Leonard CS. Function of specific K channels insustained high-frequency firing of fast-spiking neocortical interneurons.J Neurophysiol 82: 2476–2489, 1999.
Fellous JM, Rudolph M, Destexhe A, Sejnowski TJ. Synaptic backgroundnoise controls the input/output characteristics of single cells in an in vitromodel of in vivo activity. Neuroscience 122: 811–829, 2003.
Fernandez FR, White JA. Artificial synaptic conductances reduce subthresh-old oscillations and periodic firing in stellate cells of the entorhinal cortex.J Neurosci 28: 3790–3803, 2008.
French AS, Korenberg MJ. A nonlinear cascade model for action potentialencoding in an insect sensory neuron. Biophys J 55: 655–661, 1989.
French AS, Marmarelis VZ. Nonlinear neuronal mode analysis of actionpotential encoding in the cockroach tactile spine neuron. Biol Cybern 73:425–430, 1995.
French AS, Sekizawa S, Torkkeli UH. Predicting the responses of mechano-receptor neurons to physiological inputs by nonlinear system identification.Ann Biomed Eng 29: 187–194, 2001.
French AS, Wong RK. Nonlinear analysis of sensory transduction in an insectmechanoreceptor. Biol Cybern 26: 231–240, 1977.
Galarreta M, Hestrin SE. Electrical and chemical synapses among parval-bumin fast-spiking GABAergic interneurons in adult mouse neocortex. ProcNatl Acad Sci USA 99: 12438–12443, 2002.
Gamble ER, DiCaprio RA. Nonspiking and spiking proprioceptors in thecrab: white noise analysis of spiking CB-chordotonal organ afferents.J Neurophysiol 89: 1815–1825, 2003.
Gibson JR, Beierlein M, Connors BW. Functional properties of electricalsynapses between inhibitory interneurons of neocortical layer 4. J Neuro-physiol 93: 467–480, 2005.
Gilbert CD, Sigman M. Top-down influences in sensory processing. Neuron54: 677–696, 2007.
Gutkin BS, Ermentrout GB, Reyes AD. Phase–response curves give theresponses of neurons to transient inputs. J Neurophysiol 94: 1623–1635,2005.
Harvey AC. Time Series Models (2nd ed.). London: The MIT Press, 1993.Hasenstaub A, Shu Y, Haider B, Kraushaar U, Duque A, McCormick DA.
Inhibitory postsynaptic potentials carry synchronized frequency informationin active cortical networks. Neuron 47: 423–435, 2005.
Hausser M, Clark BA. Tonic synaptic inhibition modulates neuronal outputpattern and spatiotemporal synaptic integration. Neuron 19: 665–678, 1997.
Hestrin S, Galarreta M. Electrical synapses define networks of neocorticalGABAergic neurons, Trends Neurosci 28: 302–309, 2005.
Hollmann M, Heinemann S. Cloned glutamate receptors. Annu Rev Neurosci17: 31–108, 1994.
Joeken S, Schweglerand H, Richter CP. Modeling stochastic spike trainresponses of neurons: an extended Wiener series analysis of pigeon auditorynerve fibers. Biol Cybern 76: 153–162, 1997.
Juusola M, French AS. The efficiency of sensory information coding bymechanoreceptor neurons. Neuron 18: 959–968, 1997.
Kastner S, Ungerleider LG. Mechanisms of visual attention in the humancortex. Annu Rev Neurosci 23: 315–341, 2000.
1071INTEGRATION OF CONDUCTANCE INPUT IN INHIBITORY NEURONS
J Neurophysiol • VOL 101 • FEBRUARY 2009 • www.jn.org
Kawaguchi Y. Physiological subgroups of nonpyramidal cells with specificmorphological characteristics in layer II/III of rat frontal cortex. J Neurosci15: 2638–2655, 1995.
Kawaguchi Y, Kubota Y. GABAergic cell subtypes and their synapticconnections in rat frontal cortex. Cereb Cortex 7: 476–486, 1997.
Kisley MA, Gerstein GL. The continuum of operating modes for a passivemodel neuron. Neural Comput 11: 1139–1154, 1999.
Kondoh Y, Okuma J, Newland PI. Dynamics of neurons controlling move-ments of a locust hind leg: Wiener kernel analysis of the responses ofproprioceptive afferents. J Neurophysiol 73: 1829–1842, 1995.
Lien CC, Jonas P. Kv3 potassium conductance is necessary and kineticallyoptimized for high-frequency action potential generation in hippocampalinterneurons, J Neurosci 23: 2058–2068, 2003.
MacDermott AB, Dale N. Receptors, ion channels and synaptic potentialsunderlying the integrative actions of excitatory amino acids. Trends Neuro-sci 87: 280–284, 1987.
Mainen ZF, Sejnowski TJ. Reliability of spike timing in neocortical neurons.Science 268: 1503–1506, 1995.
Mancilla JG, Lewis TJ, Pinto DJ, Rinzel J, Connors BW. Synchronizationof electrically coupled pairs of inhibitory interneurons in neocortex. J Neu-rosci 27: 2058–2073, 2007.
Markram H, Toledo-Rodriguez M, Wang Y, Gupta A, Silberberg G, WuC. Interneurons of the neocortical inhibitory system. Nat Rev Neurosci 5:793–807, 2004.
Marmarelis PZ, Naka K. White-noise analysis of a neuron chain: an appli-cation of the Wiener theory. Science 175: 1276–1278, 1972.
Marmarelis VZ, Citron MC, Vivo CP. Minimum-order Wiener modelling ofspike-output systems. Biol Cybern 54: 115–123, 1986.
Matthews PB. Relationship of firing intervals of human motor units to thetrajectory of post-spike after-hyperpolarization and synaptic noise. J Physiol492: 597–628, 1996.
McCormick DA. Neurotransmitter actions in the thalamus and cerebral cortexand their role in neuromodulation of thalamocortical activity. Prog Neuro-biol 39: 337–388, 1992.
Morita K, Kalra R, Aihara K, Robinson HPC. Recurrent synaptic input andthe timing of gamma-frequency-modulated firing of pyramidal cells duringneocortical “UP” states. J Neurosci 28: 1871–1881, 2008.
Petersen CCH. The functional organization of the barrel cortex. Neuron 56:339–355, 2007.
Pillow J, Simoncelli EP. Biases in white noise analysis due to non-Poissonspike generation. Neurocomputing 52––54: 109–115, 2003.
Poliakov AV, Powers RK, Binder MD. Functional identification of theinput–output transforms of motoneurones in the rat and cat. J Physiol 504:401–424, 1997.
Poulet JFA, Petersen CCH. Internal brain state regulates membrane potentialsynchrony in barrel cortex of behaving mice. Nature 454: 881–885, 2008.
Powers RK, Binder MD. Summation of effective synaptic currents and firingrate modulation in cat spinal motoneurons. J Neurophysiol 83: 483–500,2000.
Powers RK, Dai Y, Bell BN, Percival DB, Binder MD. Contributions of theinput signal and prior activation history to the discharge behaviour of ratmotoneurones. J Physiol 562: 707–724, 2005.
Robinson HPC, Kawai N. Injection of digitally synthesized synaptic conduc-tance transients to measure the integrative properties of neurons. J NeurosciMethods 49: 157–165, 1993.
Sakmann B, Stuart G. Patch-pipette recordings from the soma, dendrites andaxon of neurons in brain slices. In: Single-Channel Recording, edited bySakmann B, Neher E. New York: Plenum, 1995, p. 199–211.
Sharp AA, O’Neil MB, Abbott LF, Marder E. Dynamic clamp: computer-generated conductances in real neurons. J Neurophysiol 69: 992–995, 1993.
Shu Y, Hasenstaub A, McCormick DA. Turning on and off recurrentbalanced cortical activity. Nature 423: 288–293, 2003.
Singer W, Gray CM. Visual feature integration and the temporal correlationhypothesis. Annu Rev Neurosci 18: 555–586, 1995.
Softky WR, Koch C. The highly irregular firing of cortical cells is inconsistentwith temporal integration of random EPSPs. J Neurosci 13: 334–350, 1993.
Svirskis G, Kotak V, Sanes DH, Rinzel J. Enhancement of signal-to-noiseratio and phase locking for small inputs by a low-threshold outward currentin auditory neurons. J Neurosci 22: 11019–11025, 2002.
Svirskis G, Kotak V, Sanes DH, Rinzel J. Sodium along with low-thresholdpotassium currents enhance coincidence detection of subthreshold noisysignals in MSO neurons. J Neurophysiol 91: 2465–2473, 2004.
Svirskis G, Rinzel J. Influence of subthreshold nonlinearities on signal-to-noise ratio and timing precision for small signals in neurons: minimal modelanalysis. Network 14: 137–150, 2003.
Tateno T, Harsch A, Robinson HPC. Threshold firing frequency–currentrelationships of neurons in rat somatosensory cortex: type 1 and type 2dynamics. J Neurophysiol 92: 2283–2299, 2004.
Tateno T, Robinson HPC. Rate coding and spike-time variability in corticalneurons with two types of threshold-dynamics. J Neurophysiol 95: 2650–2663, 2006.
Tateno T, Robinson HPC. Phase resetting curves and oscillatory stability ininterneurons of rat somatosensory cortex. Biophys J 92: 683–695, 2007.
Toledo-Rodriguez M, El Manira A, Wallen P, Svirskis G, Hounsgaard J.Cellular signalling properties in microcircuits. Trends Neurosci 28: 534–540, 2005.
Truccolo W, Eden UT, Fellows MR, Donoghue JP, Brown EN. A pointprocess framework for relating neural spiking activity to spiking history,neural ensemble, and extrinsic covariate effects. J Neurophysiol 93: 1074–1089, 2005.
Turker KS, Powers RK. Black box revisited: a technique for estimatingpostsynaptic potentials in neurons. Trends Neurosci 28: 379–386, 2005.
Wehr M, Zador AM. Balanced inhibition underlies tuning and sharpens spiketiming in auditory cortex. Nature 426: 442–446, 2003.
Westwick DT, Kearney RE. Nonparametric identification of nonlinear bio-medical systems, Part I: Theory. Crit Rev Biomed Eng 26: 153–226, 1998.
