Refractory Properties of Cochlear Implant-Induced Spiking in Auditory Nerve Fibers areDependent on Location of Stimulation and Voltage-Gated Channel Type Distribution

Jason Boulet1,2 and Ian Bruce1,2,3

Auditory Engineering Laboratory1, McMaster Integrative Neuroscience Discovery & Study2,Department of Electrical & Computer Engineering3, McMaster University, Hamilton, ON, Canada

A L

AUDITORYENGINEERINGLABORATORY

Abstract

Background Experimental work has demonstrated that auditory nerve fibers(ANFs) of cats cannot fully respond to high rates of electrical stimulation, thusreducing the information transfer to the brain. Miller et al. (2001) have shownthat a limiting factor of the reduced spike information transfer can be attributedto the neuron’s refractory period. A computational model of a node of Ranvierof the ANF (Negm and Bruce, 2008) suggested that low-threshold potassium(KLT) and hyperpolarization-activated cyclic nucleotide-gated cation (HCN)channels (Yi et al., 2010) might be responsible for a larger refractory period(Negm and Bruce, in prep.).Methods We extend that work with a simulation study taking into accountANF morphology (Woo et al., 2010) to consider the differential spiking activityas a function of 1) the location of electrical stimulation and 2) nodal channelcomposition at important locations along the ANF. Specifically, we test threeANF models variants: A) only fast Nav and delayed-rectifier Kv at all nodes,B) with the addition of KLT & HCN channels (Yi et al., 2010) at the firstperipheral node and on the nodes of Ranvier neighboring the soma and C) byexpanding the distribution of KLT channels to all nodes (Bortone et al., 2006).Results In general, we observed the absolute refractory period of model Cto be the greatest followed by model B, then by model A. Models B and Ccontrasted with model A by having a greater probability of spike initiation atthe location of stimulation. Model A did not show a strong relative refractoryperiod at its peripheral nodes. We argue that the washout of the relativerefractory period in this region was dependent on the low correlation betweenthe location of the stimulating electrode and the location of spike initiation.Conclusion Preliminary results indicate that model C is most consistent withthe published physiological data. In addition to the KLT & HCN channels ofmodel C, other ion channel types may be necessary to explain all aspects ofrefractory behavior observed in vivo.This research was supported by the Natural Sciences and Engineering ResearchCouncil of Canada (Discovery Grant #261736).

I. INTRODUCTION

I Recent studies have shown that electrically stimulated type I cat ANFsundergo drops in spike rate over the duration of a pulse train for highpulses rates (Zhang et al., 2007). Computational models of the ANF basedon the Hodgkin–Huxley equations containing only Nav and Kv channelsdo not adequately describe these decrements in spike rate.

I Miller et al. (2001) has shown that the duration of refraction can varygreatly across ANFs, which is also difficult to explain with only Nav andKv channels.

I Yi et al. (2010) have experimentally found HCN channels at the firstperipheral node (or terminal) and the nodes neighboring the soma in mousespiral ganglion cells. KLT channels have been localized on ANF axonsentering rat cochlear nucleus (Bortone et al., 2006).

I A computational membrane-node model of the cat ANF incorporatingNav, Kv, KLT and HCN channels has shown that the HCN and KLTchannels can produce increasing spike-rate adaptation and accommoda-tion with increasing stimulation rate (Negm and Bruce, 2008), as well asincreased refractory periods (Negm and Bruce, in prep.).

I We built a compartmental model of the cat ANF to better understandhow refraction depends on the location and populations of voltage-gatedion channel species (Nav, Kv, KLT and HCN) and the location and rateof electrical stimulation from a CI.

I To simulate the activity of various ion channels types, we utilize stochasticion channel models, because the resulting fluctuations in excitability arethought to be behaviorally significant for CI users (Bruce et al., 1999a,b).

II. METHODS: Ion Channel (In-)Activation and Time Constants

(In-)Activation Functions and Time Constants

0

0.25

0.5

0.75

1

Act

ivat

ion

/ Ina

ctiv

atio

n

Particlem∞ Nav Activation

h∞ Nav Inactivationn∞ Kv Activation

w∞ KLT Activationz∞ KLT Inactivation

r∞ s∞ HCN Activation

10−3

10−2

10−1

100

101

102

−150 −100 −50 0 50 100 150Relative Membrane Potential (mV)

Tim

e C

onst

ant (

ms)

Figure 1 : Steady-state activation/inactivation functions and time constants at 37◦C. Weadjusted w∞, z∞, r∞ and s∞ from 22◦C to 37◦C (Cartee, 2000; Rothman and Manis, 2003).

II. METHODS: ANF Models

Compartmental Model

· · ·p1 p2 p3 p4 c1 c2 c3 c4 c23

10 150 150 150 32.6 150 200 250 300 350

1

1.2

2.3

Peripheral axon Central axonSoma unit: µm

Figure 2 : Feline ANF morphology is based on Woo et al. (2010). The soma is myelinated,which contrasts with the mouse and human ANF.

Ion Channel Distribution

0

50

100

0

50

100

0

50

100

AB

C

p1 p2 p3 p4 c1 c2 c3 c4 c5Node of Ranvier

Cha

nnel

Den

sity

(µm

−2)

ChannelNaKKLTHCN

Figure 3 : Hossain et al. (2005) found high densities of Nav1.6 channels located at p1, p4and c1 in the mouse ANF. Yi et al. (2010) have shown HCN channels at the same nodesin mouse spiral ganglion cells. KLT channels have been localized on ANF axons entering ratcochlear nucleus (Bortone et al., 2006).

Circuit Model

· · ·

Ra,(k−2,k−1)

Cm,k−

1

Rm,k−

1

Elea

k,k−

1

Ve,k−1

Ra,(k−1,k)

Cm,k

Rm,k

Elea

k,k

gNa,k

ENa

gK,k

EK

gKLT

,k

EKLT

gh,k

Eh

Ve,k

Ra,(k,k+1)Cm,k+1 Rm,k+1

Elea

k,k+1

Ve,k+1

Ra,(k+1,k+2)

· · ·

Ra,(k+8,k+9)

Cm,k+9 Rm,k+9

Elea

k,k+9

Ve,k+9

Ra,(k+9,k+10)

· · ·

Node of Ranvier1 of 9 Myelin 9 of 9 Myelin

Figure 4 : The circuit model is solved as a discretized (∆t = 2.5 µs) partial differentialequation (Mino et al., 2004) via time-centered Crank–Nicholson, also used in the neuralsimulation environment NEURON (Carnevale and Hines, 2006).

II. METHODS: Ion Channel Simulation

Channel kinetics obey continuous-time discrete-state Markov processes. Thestate transition diagrams for Nav, Kv (Mino et al., 2002), KLT (Negm andBruce, 2008) and HCN1,4 (Liu and Davis, 2012) are shown in Table 1. Redstates indicate fully open states that contribute to conducting ionic current.We simulated the 4 voltage-gated ion channel types with a channel numbertracking algorithm (Chow and White, 1996; Mino et al., 2002).

Table 1 : State transition diagrams

Nav Kv

m0h0

3αm

βm

m1h0

2αm

2βm

m2h0

αm

3βm

m3h0

αh��βh αh��βh αh��βh αh��βh

m0h1

3αm

βm

m1h1

2αm

2βm

m2h1

αm

3βm

m3h1

n0

4αn

βn

n1

3αn

2βn

n2

2αn

3βn

n3

αn

4βn

n4

KLT HCN

w0z0

4αw

βw

w1z0

3αw

2βw

w2z0

2αw

3βw

w3z0

αw

4βw

w4z0

αz��βz αz��βz αz��βz αz��βz αz��βz

w0z1

4αw

βw

w1z1

3αw

2βw

w2z1

2αw

3βw

w3z1

αw

4βw

w4z1

r0

2αr

βr

r1

αr

2βr

r2

s0

αs

βs

s1

II. METHODS: Electrical Stimulation

Monophasic

Biphasic

unit: µs 25 25IPI

I Three distances from the ANF: intracellular, 0.5 and 1 mmI Nine sites of stimulation, over nodes p1 to c5I Monophasic (cathodic) and biphasic (cathodic, then anodic)I Spherical monopolar extracellular electrode radius: 150 µmI Inter Pulse Intervals (IPIs) ranging from 0.3 to 10 msI 1st pulse amplitude: I1 = (1 + 3RS) θ, 2nd pulse: variableI Successfully propagated spike is voltage discriminated at c17

III. RESULTS: Single-Pulse Response

Firing Efficiency (FE)

σ0

0.5

1

340 360 380 400Injected Current (µA)

Firi

ng E

ffici

ency

, FE

Figure 5 : FE is the probability of aspike given a single pulse of current in-put. Data points are the mean valuesfrom 1000 simulation trials and arefit to an integrated Gaussian (Bruceet al., 1999a). The threshold currentθ is the current at FE = 0.5 and therelative spread is RS = σ/θ. Thisexample is the response from an elec-trode placed 0.5 mm above node c3.

Spatial Spiking ResponseMono Bi

c5c4c3c2c1p4p3p2p1

c5c4c3c2c1p4p3p2p1

c5c4c3c2c1p4p3p2p1

AB

C

p1 p2 p3 p4 c1 c2 c3 c4 c5 p1 p2 p3 p4 c1 c2 c3 c4 c5

Site of Stimulation

Spi

ke In

itiat

ion

Nod

e

0.1

0.2

0.3

0.4

0.5

Probabilityof spikeinitiation

0.900

0.925

0.950

0.975

0.7

0.8

0.9

1.0

0.25

0.50

0.75

Intracellular0.5 m

m1 m

m

0 0.25 0.5 0.75 1Firing Efficiency, FE

Spe

arm

an's

ρ StimMonoBi

ModelABC

Figure 6 : (left) Probability of spike initiation from single pulse stimulation. Results weregathered from 1000 simulation trials. Probability is mapped onto a color as function of thestimulated site (x-axis) and the spike initiation node (y -axis). Gray values indicate no spike.This figure shows the response for an electrode 0.5 mm away from the ANF at FE = 0.5.(right) Spearman’s correlation (ρ). This is a summary of the plot on the left, but over allFE’s and distances.

Threshold Current and Relative Spread

7.18 × 10−5

1.34 × 10−4

1.96 × 10−4

2.58 × 10−4

4.04 × 102

5.84 × 102

7.65 × 102

9.45 × 102

2.22 × 103

2.80 × 103

3.37 × 103

3.95 × 103

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

Thr

esho

ld C

urre

nt, θ

(µA

)

0.03

0.04

0.05

0.06

0.07

0.03

0.04

0.05

0.06

0.025

0.030

0.035

0.040

0.045

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

Rel

ativ

e S

prea

d, R

S

StimMonoBi

ModelABC

Figure 7 : (left) Threshold current (θ) and (right) Relative Spread (RS) across all nodesover 1000 trials.

Mean Latency at FE = 0.5

40

60

80

100

120

60

80

100

120

80

100

120

140

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

Initi

atio

n La

tenc

y (µ

s)

400

500

600

400

500

600

400

450

500

550

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

Pro

paga

tion

Del

ay (

µs)

StimMonoBi

ModelABC

Figure 8 : Mean Latency at FE = 0.5 over 1000 trials. (left) Initiation latency and (right)propagation delay. Both components sum to give the time the spike takes to arrive at nodec17.

Jitter at FE = 0.5

10

20

30

10

20

30

40

20

30

40

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

Initi

atio

n Ji

tter

(µs)

10

20

30

10

20

30

40

1020304050

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

Pro

paga

tion

Del

ay J

itter

(µs

)

StimMonoBi

ModelABC

Figure 9 : Jitter at FE = 0.5 for 1000 trials. (left) Initiation latency and (right) propagationdelay jitter.

III. RESULTS: Spike Initiation and Propagation

Relative Membrane PotentialModel A Model B Model C

c23c22c21c20c19c18c17c16c15c14c13c12c11c10c9c8c7c6c5c4c3c2c1p4p3p2p1

c23c22c21c20c19c18c17c16c15c14c13c12c11c10c9c8c7c6c5c4c3c2c1p4p3p2p1

Mono

Bi

0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2Time (ms)

Nod

es o

f Ran

vier

140

120

100

80

60

40

20

0

−20

−40

V (mV)

Figure 10 : One example of a simulation of the relative membrane potential as a function oftime and node of Ranvier along the ANF. For these particular trials, we present the ANF witha stimulus 0.5 mm over node p4, with an IPI of 750 µs and a second-pulse magnitude of 1.5θ.

III. RESULTS: Threshold Ratio (2nd-pulse/1st-pulse)

We fit the threshold data to

θ2/1 =a1 + a2

a1

(1− exp

[−(

IPI−τabsτ1

)])+ a2

(1− exp

[−(

IPI−τabsτ2

)]) (1)

which has been found to fit the threshold ratio data better with two time scales(Negm and Bruce, in prep.) than with one (Miller et al., 2001).

Threshold RatioMono Bi

1.0

1.5

2.0

2.5

0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0Inter Pulse Interval (ms)

Thr

esho

ld C

urre

nt R

atio

, θ2

1

ModelABC

Figure 11 : Second-to-first pulse threshold ratio (θ2/1). Second pulse thresholds are derivedfrom 100 simulation trials fit to an integrated Gaussian function. In these cases above, weused a stimulus 0.5 mm over node c3.

III. RESULTS: Time Scales of Refraction

Relative Refractory Periods

10−510−410−310−210−1100

10−4

10−3

10−2

10−1

10−5

10−4

10−3

10−2

10−1

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

τ 1 (

ms)

10−0.2

10−0.1

100

100.1

100.2

10−0.2

100

100.2

100.4

10−0.2

10−0.1

100

100.1

100.2

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

τ 2 (

ms)

StimMonoBi

ModelABC

Figure 12 : Relative refractory periods (left) τ1 and (right) τ2.

Absolute Refractory Period

0.3

0.6

0.9

1.2

0.2

0.4

0.6

0.8

0.4

0.6

0.8

Intracellular0.5 m

m1 m

m

p1 p2 p3 p4 c1 c2 c3 c4 c5Site of Stimulation

τ abs

(m

s)

StimMonoBi

ModelABC

0.3

0.6

0.9

1.2

Mono BiStim

τ abs

(m

s) ModelABC

Figure 13 : Absolute refractory period τabs (left) arranged by site of stimulation, distance,model and stimulus, whereas (right) we also show the median values collapsed across site ofstimulation and distance.

III. RESULTS: Spike-Rate Adaptation

Post-Stimulus Time-Histograms at FE = 0.5p1 p2 p3 p4 c1 c2 c3 c4 c5

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0.00

0.25

0.50

0.75

1.00

0.00

0.25

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1.00

0.00

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0.50

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1.00

0.00

0.25

0.50

0.75

1.00

0.2 kHz

0.8 kHz

2 kHz

5 kHz

0 50 100

150

200 0 50 100

150

200 0 50 100

150

200 0 50 100

150

200 0 50 100

150

200 0 50 100

150

200 0 50 100

150

200 0 50 100

150

200 0 50 100

150

200

Time (ms)

Firi

ng R

ate

(kH

z)

Model●

●

●

ABC

Figure 14 : PSTHs for our model ANFs. The responses are shown for stimulated nodes p1to c5 at the stimulus rates 200, 800, 2000 and 5000 kHz with monophasic stimulation and ata distance of 0.5 mm. The PSTHs were generated by averaging across 100 simulation trialsof 200 ms for two sets of time-bins (Zhang et al., 2007).

IV. CONCLUSIONS

I Single-pulse threshold currents in models C>B>A.I Models B and C show a stronger relationship between where the stimulus

is delivered and where the spike is initiated than model AI Models B and C show the best fits to the two-time scale refractory func-

tion, as evidenced by the unwieldy relative refractory time constants ofmodel A.

I Model C has a significantly larger absolute refractory period than modelsA and B. Therefore, cell-wide distribution of KLT channels plays a majorrole in increasing the absolute refractory period.

I Model A displays evidence of summation for high stimulus rates of 2000and 5000 pulses/s, similarly to Heffer et al. (2010) in guinea pig ANF.

I Further computational studies must be done to address the relative im-pacts of refraction, spike-rate adaptation, accommodation and facilitation(summation) on changes in spike rate over the duration of a pulse trainfor high stimulation rates.

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