PEAK-SPLITTING IN THE RESPONSE OF THE LEAKY

INTEGRATE-AND-FIRE NEURON MODEL TO LOW-FREQUENCY

PERIODIC INPUTS
Levin Kuhlmann1,2,∗ , Anthony Burkitt2 , and Graeme M. Clark1,2
1
Department of Otolaryngology, The University of Melbourne, Royal Victorian Eye and
Ear Hospital, 32 Gisborne Street, East Melbourne, VIC 3002, Australia
2
The Bionic Ear Institute, 384-388 Albert Street, East Melbourne, VIC 3002, Australia
∗
l.kuhlmann@medoto.unimelb.edu.au

Abstract - The cause of peak-splitting in frequency responses will show similar peak-splitting
the output phase distribution of the leaky to that seen in IHC responses. It is theoretically pos-
integrate-and-fire single neuron model in re- sible, however, for AN fibres to show peak-splitting
sponse to low-frequency periodic nerve fibre in their low-frequency responses even when there is
inputs is analyzed. It is found that peak- no peak-splitting present in the attached IHC low-
splitting largely arises from an increase of the frequency responses. If this were to occur, then it
spiking-rate of individual nerve fibre inputs, is likely to arise as a result of non-linearities asso-
or from an increase in amplitude of individual ciated with the way in which a AN fibre processes
input excitatory postsynaptic potentials, or its IHC input. Cai and Geisler [1] demonstrated
both. These findings add another dimension that peak-splitting in AN fibre responses was unpre-
to the understanding of how peak-splitting dictable, thus giving strong indication of non-linear
arises in the phase histograms of the responses effects. Cochlear nucleus (CN) neurons, which re-
of neurons in the auditory pathway, given that ceive inputs from AN fibres, are also thought to
peak-splitting is typically thought to arise as a demonstrate peak-splitting in their response to low-
result of the non-linear dynamics of the basi- frequency tones. The goal of the present study is to
lar membrane and the hair cells. This research understand how peak-splitting in the low-frequency
has implications for the understanding of the responses of CN neurons depends on neuronal param-
temporal code in the auditory pathway. eters, such as the spiking-rate of individual input AN
Index Terms - auditory pathway, leaky fibres and the amplitude of the input excitatory post-
integrate-and-fire neuron model, peak-splitting, synaptic potentials (EPSPs). To achieve this goal a
temporal coding leaky integrate-and-fire (LIF) single neuron model of
a CN neuron receiving stochastic periodic AN fibre
I. INTRODUCTION inputs was implemented. In this model input EPSPs
to the neuron are summed and an action potential
When stimulating auditory nerve (AN) fibres with
(AP or spike) is generated when the membrane po-
a low-frequency (≤ 1 kHz) tonic sound stimulus, in
tential reaches threshold.
most cases there is only a single peak in a phase
histogram constructed on the period of the low-
frequency tone [1]. In some cases, however, when II. METHODS
the stimulus intensity is increased, two or even three
peaks can be seen in the phase histogram [1]. Peak- II.1. Neural Model
splitting is the term given to this peak multiplication
that occurs within a phase histogram when there is an The analysis presented here considers a single neu-
increase in intensity of a low-frequency tonic sound ron with N independent inputs (afferent fibres) as-
stimulus [1], [2], [3]. The work done on peak-splitting sumed to have the same synaptic response amplitude,
in the auditory pathway has concentrated on inner a, and time course, s(t). The time course of an in-
hair cells (IHCs) of the cochlear [4], [5] and AN fi- put at the site of spike generation is described by the
bres [1], [2], [3], [4], [6], [7], [8], [9]. synaptic response function u(t) for the LIF neuron
An understanding of the causes of peak-splitting in model. The membrane potential is assumed to be re-
these cell types is slowly being elucidated. Cody and set to its initial value at time t = 0, V (0) = v0 , after
Mountain [5] have demonstrated that peak-splitting an AP has been generated. An AP is produced only
in low-frequency responses of IHCs arises from the when the membrane potential exceeds the threshold,
mechanical input to the IHC, i.e. the vibrations of Vth , which has a potential difference with the reset
the basilar membrane (BM) relative to the tectorial potential of θ = Vth − v0 . After an AP has been
membrane. Furthermore, the IHCs provide the in- generated there is an absolute refractory period, τr ,
puts to AN fibres and so it is clear that AN fibre low- during which no APs can be generated. The mem-
brane potential is the sum of the input EPSPs by Burkitt and Clark [13], [14], [15]. Based on a de-
∞
scription of the membrane potential and the synaptic
X inputs, one can now temporally integrate the synap-
V (t) = v0 + N a s(t), s(t) = u(t − tm ), (1)
tic inputs and the probability density of the mem-
m=1
brane potential can be obtained in the Gaussian ap-
where the index m denotes the mth input AP from proximation. Using a generalization of the renewal
the particular fibre, whose time of arrival is tm (0 < equation, the probability density of the membrane
t1 < t2 < ... < tm < ...). The rate of the input potential and the probability density of the mem-
AP arrival times is discussed in the following section. brane potential conditional on the initial phase of
The synaptic response function, u(t), is the inputs can be used to evaluate the conditional
first-passage time density (equivalent to the inter-
 −t/τ
e for t ≥ 0 spike interval distribution conditional on the initial
u(t) = (2) phase of the inputs). The output of the model, the
0 for t < 0,
phase distribution (synonymous with the phase his-
where τ is the decay time constant of the membrane togram), can then be evaluated. The phase distribu-
potential. Consequently the membrane potential has tion is the stationary solution to a phase transition
a discontinuous jump of size a upon the arrival of an density defined by the periodic wrapping of the con-
EPSP and then decays exponentially between inputs. ditional first-passage time densities [15], [16]. As a
The decay of the EPSP across the membrane means result of obtaining the phase distribution, the SI of
that the contribution from EPSPs that arrive earlier the model’s output can be evaluated numerically [10].
have partially decayed by the time that later EPSPs
arrive. In this study the voltage scale is set so that III. RESULTS
v0 = 0. Peak-splitting was observable in the output phase
distributions of the LIF neuron model. It arose as a
II.2. Synaptic Input result of an increase in the average spiking-rate per
In this study it is assumed that the input spiking- period of the input, p. Fig. 1 demonstrates the ap-
rate on each of the input fibres is identical. The time- pearance of peak-splitting in the output phase dis-
dependent rate of arrival of input spikes at a synapse tribution as a result of this increase of average input
is periodic with period T and initial phase tφ , spiking-rate. The ordinate represents the spike phase
density, χ(tφ ) (where tφ is the phase at which an AP
∞ !
is produced), and the abcissa represents the time for

X 1 (t − kT + tφ )2
λ(t) = p √ exp − , (3)
2πσ 2 2σ 2 one stimulus period. All model parameter values are
k=−∞
given in the figure caption. In Fig. 1(a) and (b) the
where, for a single fibre, p is the time-averaged input average spiking-rate per input is λin = 300 and 800
spiking-rate per period and σ is the standard devi- spikes/s respectively (i.e. p = 1 and 4 spikes/period),
ation (S.D.) of the Gaussians. The synchronization the output SI is Sout = 0.95 and 0.57 respectively and
index (SI - a measure of the degree of locking to a the average output spiking-rate is λout = 50.81 and
phase of the period of the input [10]) of the above 282.04 spikes/s respectively. As the input spiking-
spiking-rate function, denoted S, can be calculated rate increases, the phase distribution spreads to the
as follows [11], right and eventually the original peak splits into two,
with the two resulting peaks being 1.61 ms apart.
Peak-splitting was also found to occur by increas-
 2 2

2π σ
S = exp − . (4) ing EPSP amplitude, a, while keeping the average
T2
spiking-rate per input fixed. Fig. 2 demonstrates
This expression for S can be calculated by dividing the appearance of peak-splitting in the output phase
the first complex Fourier coefficient of λ(t) by the ze- distribution as a result of an increase in a. The or-
roeth complex Fourier coefficient of λ(t). The period, dinate represents the spike phase density, χ(tφ ), and
T , and the S.D. of the Gaussians, σ, can be varied in- the abcissa represents the time for one stimulus pe-
dependently to provide the desired S value. The use riod. All model parameter values are given in the
of the term “period” (or “frequency” f = T1 ) here and figure caption. For Fig. 2(a) and (b) the EPSP am-
throughout the analysis refers to this periodic modu- plitude is a = 0.05 and 0.25 respectively, the output
lation of the spiking-rate of the inputs. Furthermore, SI is Sout = 0.95 and 0.55 respectively and the av-
for the sake of using conventional terms, the average erage output spiking-rate is λout = 50.81 and 531.39
spiking-rate per input can be defined as λin = Tp . spikes/s respectively. As the EPSP amplitude in-
This input spiking-rate function, Equation (3), rep- creases, the phase distribution spreads to the right
resents an inhomogeneous Poisson process [12]. and eventually the original peak splits into two, with
the two resulting peaks being 1.24 ms apart.
II.3. Output of the Model
Now that the neural model and its synaptic inputs IV. DISCUSSION
have been defined, the output of the model can be As was demonstrated in Fig. 1 and Fig. 2 peak-
evaluated by using the following methods developed splitting in the output of the model was shown to
 λin = 300 spikes/s
1500
Spike phase density χ(tφ)

1000

500

0
0 2.5 5
Time for one stimulus period (ms)

(a) (a)

λin = 800 spikes/s

1000

500

0
0 2.5 5
Time for one stimulus period (ms)

(b) (b)

Fig. 1. Plots of phase distributions demonstrating the appear- Fig. 2. Plots of phase distributions demonstrating the appear-
ance of peak-splitting as a result of an increase in average ance of peak-splitting in the output phase distribution as
spiking-rate per input. The ordinate represents the spike a result of an increase in the EPSP amplitude, a. The or-
phase density, χ(tφ ), and the abcissa represents the time dinate represents the spike phase density, χ(tφ ), and the
for one stimulus period, measured in milliseconds (ms). abcissa represents the time for one stimulus period, mea-
The neural parameter values are, N = 20 inputs, a = 0.05, sured in milliseconds (ms). The neural parameter values
θ = 1, τr = 1 ms and τ = 2 ms. The input parameter val- are, N = 20 inputs, θ = 1, τr = 1 ms and τ = 2 ms. The
ues are, T = 5 ms and Sin = 0.5. For (a) and (b) the av- input parameter values are, T = 5 ms, λin = 300 spikes/s
erage spiking-rate per input is λin = 300 and 800 spikes/s (p = 1.5 spikes/period) and Sin = 0.5. For (a) and (b)
respectively (i.e. p = 1.5 and 4 spikes/period), the output the EPSP amplitude is a = 0.05 and 0.25 respectively, the
SI is Sout = 0.92 and 0.57 respectively and the average output SI is Sout = 0.95 and 0.55 respectively and the
output spiking-rate is λout = 50.81 and 282.04 spikes/s average output spiking-rate is λout = 50.81 and 531.39
respectively. spikes/s respectively.

occur as a result of an increase in the average spiking- more inputs than in Fig. 1 and the spiking-rate of
rate per input, λin , and an increase in the EPSP am- individual inputs increases in a realistic manner such
plitude of the inputs, a, respectively. An increase that the total number of input spikes per second is
in the average spiking-rate per input, λin , effectively the same as in Fig. 1. In Fig. 2, where the oc-
corresponds to an increase in stimulus intensity. As currence of peak-splitting is demonstrated for an in-
is shown in Fig. 1 for a stimulus frequency of 200 Hz, crease in EPSP amplitude, seemingly realistic values
a seemingly unrealistic value for the average spiking- of the EPSP amplitude relative to the size of the cell’s
rate per input is required in order for peak-splitting threshold are required for peak-splitting to occur. In
to occur. In Fig. 1 an average spiking-rate per input Fig. 2 the value of a for which peak splitting occurs
of λin = 800 spikes/s is required for peak-splitting is one quarter of the threshold. Considering that the
to occur. Clearly, as a result of the typical refrac- neural model represents a CN neuron, such a value
tory period of a neuron (of the order of 1 ms (see for the EPSP amplitude appears to be realistic [18],
[17] for a review), such a value for the average in- [19].
put spiking-rate does not appear realistic. However, The height, shape and number of the peaks in the
given the design of the LIF neuron model presented phase distribution depends on several of the param-
here, the total number of input spikes per second, i.e. eters of the model. The height and shape of each
N × λin , ultimately determines the output response. peak depends largely on the parameters that deter-
Thus it is possible that the peak-split response seen mine the SI of the output (results not presented here).
in Fig. 1 may arise in a situation where there are The number of peaks depends on the average input
spiking-rate and input EPSP amplitude, as is dis- [8] Joris, P. X., & Yin, T. C. T. 1992. Responses to amplitude-
cussed in this section, where an increase in the val- modulated tones in the auditory nerve of the cat. J. Acoust.
Soc. Am., 91:1, 215–232.
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Detection with Noisy Periodic Spike Input. Neural Comput.,
As was discussed in the introduction, peak- 10, 1987–2017.
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Stochastic Processes. Singapore: McGraw-Hill International
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Input and Spike Output in Neural Systems. Neural Com-
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IHCs and AN fibres. In the present study, it seems [14] Burkitt, A. N., & Clark, G. M. 2000a. Calculation of In-
clear that peak-splitting can be generated in a CN terspike Intervals for Integrate and Fire Neurons with Pois-
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this situation, but also in the situation where there is [15] Burkitt, A. N., & Clark, G. M. 2000b. Synchronization
of the Neural Response to Noisy Periodic Synaptic Input.
no peak-splitting in BM vibrations or the input AN fi- Manuscript submitted.
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Much is known about the way that components
of the auditory pathway code sounds in the times of
their responses. However, the role that peak-splitting
plays in temporal coding in the auditory pathway is
not quite clear. It is hoped that future studies will
help to elucidate this role.

VI. ACKNOWLEDGEMENTS
The authors would like to thank A. Paolini and B.
Uragun for their useful comments on the manuscript.

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