Neurobiology 735 - third module
• Week 1: Coding
• Week 2: Neurons and synapses
• Week 3: Networks
• Week 4: Learning and memory
Textbook: Dayan and Abbott
Coding 1:
Models of neuronal responses to external stimuli
Response of neurons to sensory stimuli
• Average firing rate across trials as a function of time: Post-Stimulus Time Histogram (PSTH)
• Average firing rate in a temporal epoch before the stimulus onset (background rate) or after the
stimulus onset (sensory response)
• Variability across time: Coefficient of Variation (CV) = SD(ISI)/Mean(ISI)
• Variability across trials: Fano Factor (FF) = Var(Spike count)/Mean
CVs of cortical neurons
PFC
Softky and Koch 1993 Compte et al 2003
FFs of cortical neurons
Churchland et al 2010
Modeling spike trains as point processes
I I II I I I
• Renewal processes: each interspike interval is drawn independently from previous
intervals, from a p.d.f. ρ(T ).
Examples of ISI distributions used in neuroscience:
– Exponential (Poisson process)
– Gamma
– Inverse Gaussian
Example 1: Poisson
• pdf of ISIs
P (T ) = ν exp(−νT )
• mean
1/〈T 〉 = ν
• CVSD(T )
〈T 〉= 1
• Spike counts in disjoint intervals are in-
dependent.
• Spike count in an interval of duration T
is given by a Poisson distribution,
P (k) = (νT )k exp(−νT )/k!
0 1 2 3 4 5Interspike Interval (ISI)/Mean
0
0.2
0.4
0.6
0.8
1
Pro
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• Mean spike count in interval of duration
T : νT
• Variance of spike counts is equal to the
mean
• Fano Factor FF=1
Example 2: Gamma
• pdf of ISIs
P (T ) =(kν)k
Γ(k)T k−1 exp(−kνT )
• mean
1/〈T 〉 = ν
• CVSD(T )
〈T 〉=
1√k
0 1 2 3 4 5Interspike Interval (ISI)/Mean
0
0.2
0.4
0.6
0.8
1
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k=0.5
k=1
k=2
k=5
Example 3: Inverse Gaussian
• pdf of ISIs
P (T ) =
√k
2πT 3νexp
(−k(νT − 1)2
2Tν
)• mean
1/〈T 〉 = ν
• CVSD(T )
〈T 〉=
1√k
0 1 2 3 4 5Interspike Interval (ISI)/Mean
0
0.5
1
1.5
Pro
ba
bili
ty d
en
sity
k=0.5
k=1
k=2
k=5
How do spike trains depend on external stimuli?
1. Mean rates
Continuous stimuli
Tuning curves quantify how the firing rate of a neuron depends on a continuous parameter
characterizing the stimulus.
• Bell-shaped tuning curves
– Orientation selectivity in V1;
– Direction selectivity in MT;
– Spatial location of the animal in HPC of the rat;
– Spatial location of stimulus in PPC, PFC;
– Location of a saccade in FEF;
– Direction of the arm in M1;
– Head direction in DTN, thalamus, subiculum;
• Monotonic tuning curves
– Eye position in oculomotor nuclei
– Angular velocity of the head in vestibular nuclei
– Frequency of vibration in S1, S2, PFC
– Retinal disparity in V1
V1: orientation
r = r0 + (rmax − r0) exp
(−1
2
(s− smax
σ
)2)
Hubel and Wiesel 1968
M1: arm direction
r = r0 + (rmax − r0) cos(s− smax)
Georgopoulos et al 1982
Auditory system: frequency
Bartlett et al 2011
Prefrontal cortex
r = r0 + (rmax − r0) exp
(−1
2
(s− smax
σ
)2)
Funahashi et al 1989
Head-direction cells
Taube 1995
Oculomotor nuclei: eye position
Aksay et al 2000
Discrete stimuli
• Olfactory system: odors
• Temporal lobe: objects
Typically, neurons respond only to a small fraction of all possible stimuli (sparse coding)
Rat olfactory cortex - odors
Poo and Isaacson 2009
Primate IT cortex - objects
Woloszyn and Sheinberg 2012
Human hippocampus - people
Quiroga et al 2005
How are dynamic stimuli encoded by single neurons?
Characterizing input/output transformation: Volterra/Wiener series
• How to characterize the transformation from a time-dependent input s(t) into a
time-dependent neuronal output r(t)?
• Volterra/Wiener series:
r(t) = F [s(t)] = g0 +
∫dt1g1(t1)s(t− t1)
+
∫dt1dt2g2(t1, t2)s(t− t1)s(t− t2)
+ . . .
• gn = Volterra/Wiener kernels
Wiener series
• Wiener approach:
– Use a stochastic signal s(t) (simplest case: white noise)
– In the Wiener expansion, individual terms are statistically independent
Approximating white noise
• White noise = mathematical idealization
• No physical system can generate truly white
noise
• Approximation: generate noise at discrete time
steps, t = m∆t where m is an integer, ∆t
should be much smaller than the characteristic
time scales of the system;
• At each time step, generate s from a Gaussian
pdf with mean zero and variance S/∆t, inde-
pendently from previous steps
Computing Wiener kernels
• Use white noise stimulus s(t)
• Record spike train y(t) in response to the stimulus:
I I II I I I
• Correlate output with products of white noise input at different times::
g0 = 〈y(t)〉g1(τ) = 〈y(t)s(t− τ)〉
g2(τ1, τ2) = 〈y(t)s(t− τ1)s(t− τ2)〉. . .
• g0 = mean firing rate
• g1(τ) = Spike-Triggered Average of the stimulus
• g2(τ1, τ2) = Spike-Triggered Covariance of the stimulus
• . . .
The spike-triggered average (STA)
• Stimulus s(t)
• Measure spike train
y(t) =
n∑i=1
δ(t− ti)
• Spike triggered average (STA)
g1(τ) =1
n
n∑i=1
s(ti − τ)
=
∫s(t− τ)y(t)dt∫
y(t)dt
Visual system: Spatio-temporal receptive fields
• Spatio-temporal white-noise stimulus
s(x, y, t)
• Compute first-order Wiener kernel
g(x, y, τ) using STA
• Receptive field = {x, y} for which
g(x, y, τ) 6= 0
• Separable receptive field:
g(x, y, τ) = gs(x, y)gt(τ)
• Non-separable receptive field:
g(x, y, τ) 6= gs(x, y)gt(τ)
RFs in thalamus and retina
• Circular central ON (OFF) surrounded by annu-
lar OFF (ON)
• Fitted by a difference of Gaussians:
gs(x, y) = ±(
1
2πσ2c
exp
(−x
2 + y2
2σ2c
)− B
2πσ2s
exp
(−x
2 + y2
2σ2s
))– B = balance between center and surround
– σc = width of center
– σs = width of surround
Spatial RF of a simple cell in V1
Fitted by a Gabor function:
gs(x, y) = exp
(−x′2 + γ2y′2
2σ2
)cos
(2πx′
λ+ ψ
)x′ = x cos(θ) + y sin(θ)
y′ = −x sin(θ) + y cos(θ)
• λ = wavelength
• θ = orientation
• ψ = phase offset
• σ = width of Gaussian envelope
• γ = spatial aspect ratio
Temporal evolution of a spatial receptive field
Temporal structure of a RF
LNP model
• Can we account for non-linearities in a simple way?
• Linear-Nonlinear-Poisson (LNP) neuron model:
– Instantaneous firing rate given by
r(t) = Φ
(∫ t
−∞g1(t− t′)s(t′)dt′
)characterized by temporal filter (kernel) g1, and static non-linearity Φ
– Spike train = Poisson process with instantaneous firing rate r(t)
Fitting a LNP model to data
• Instantaneous firing rate given by
r(t) = Φ
(∫ t
−∞g1(t− t′)s(t′)dt′
)• Compute temporal filter g1 from the data
using STA;
• Then plot r(t) as a function of
L =
∫ t
−∞g1(t− t′)s(t′)dt′
• Compute Φ as the average of the data
points, and (optional) fit it with a particular
function.
Example: fitting retinal ganglion cell data (Pillow et al 2005)
Model vs data
Bibliography
• Dayan and Abbott, “Theoretical Neuroscience: Computational and Mathematical
Modeling of Neural Systems” (MIT Press, 2001), chapters 1& 2