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3. Spiking neurons and
response variability
Fundamentals of Computational Neuroscience, T. P. Trappenberg, 2002.
Lecture Notes on Brain and Computation
Byoung-Tak Zhang
Biointelligence Laboratory
School of Computer Science and Engineering
Graduate Programs in Cognitive Science, Brain Science and Bioinformatics
Brain-Mind-Behavior Concentration Program
Seoul National University
E-mail: [email protected]
This material is available online at http://bi.snu.ac.kr/
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Outline
3.1
3.2
3.3
3.4
Integrate-and-fire neurons
The spike-response model
Spike time variability
Noise models for IF-neurons
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3.1 Integrate-and-fire neurons
3.1.1 Stereotyped spike forms

Conductance-based model is too heavy to a large
network simulation
 Integrate-and-fire neuron model
 The form of spike generated by neuron is very stereotyped.
The precise form of the spike does not carry
information.
The occurrence of spikes is important.
 The relevance of the timing of the spike for information
transmission.
 Neglect the detailed ion-channel dynamics.
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3.1.2 The basic integrate-and-fire neuron
du (t )
 u (t )  RI (t ) (leaky itegrator)
dt
(3.2) I (t )   w j (t  t jf )
(3.1)
m
Membrane potential, u
j t
 Membrane time constant,  m
α  function : f ( x)  x exp(  x)
 Input current, I (t )
(3.3) u (t f )  
 Synaptic efficiency, w j
(3.4) lim u (t f   )  u res
 Firing time of presynaptic neuron  0
of synapse j, t jf
 Firing time of the postsynaptic
neuron, u (t f )
 Firing threshold, 
 Reset membrane potential, ures

f
j
 Absolute refractory time by
holding this value
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Fig. 3.1 Schematic illustration of a leaky integrate-and-fire
neuron. This neuron model integrates(sums) the external
input, with each channel weighted with a corresponding
synaptic weighting factors wi, and produces an output spike
if the membrane potential reaches a firing threshold.
3.1.3 Response of IF neurons to constant
input current (1)



Simple homogeneous differential equation,
du (t )
 Initial membrane potential 0
m
 u (t )  0
dt
 u(t=0)=1. very short input pulse.
(3.5)
 Equilibrium equation of the membrane potential after a constant
current has been applied for a long time u(t )  e t / m (3.6)
IF-neuron driven by a constant input current du
u  RI
0
 Low enough to prevent the firing.
(3.7)
ut
 After some transient time, the membrane potential dose not change
(3.8)
The differential equation for constant input (current) for all times after the
constant current Iext = const is applied:
u (t )  RI (1  e t / m 
u (t  0) t / m
e
)
(3.9)
RI
 Exponential decay of potential at u(t=0)
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3.1.3 Response of IF neurons to constant
input current (2)
RI  
RI  
Fig. 3.2 Simulated spike trains and membrane potential of a leaky integrate-and-fire neuron. The
threshold is set at 10 and indicated as a dashed line. (A) Constant input current of strength RI = 8,
which is too small to elicit a spike. (B) Constant input current of strength RI = 12, strong enough
to elicit spikes in regular intervals. Note that we did not include the form of the spike itself in the
figure but simply reset the membrane potential while indicating that a spike occurred by plotting
a dot in the upper figure.
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3.1.4 Gain function (activation function)

The time tf is given by the time when the membrane reaches
the firing thresholdu(t )   , t   ln u RIRI (3.10)
Activation or gain function define as the inverse of tf or the
firing rate r  (t   ln u RIRI ) (3.11)
f
f
m
res

1
ref
m
res
 Absolute refractory time t ref

This function quickly reaches
an asymptotic linear behavior
 A threshold-linear function is
often used to approximate
the gain function of IF-neurons
Fig. 3.3 Gain function of a leaky integrateand-fire neuron for several values of the
reset potential ures and refractory time tref.
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3.2 The spike-response model (1)

An arbitrary external current stream, I (t )
 More recent spikes have a larger influence on the membrane
potential than more distant spikes. u(t )  R es / I (t  s)ds (3.12)
m
0
 sum over all the exponential responses to very short current pulse

The spike-response model
u (t )   w j ε (t  t f , t  t jf )   (t  t f )
j t
t
(3.13)
 tf: last postsynaptic spike
 tjf: individual presynaptic spikes
 ε: The response (change) in the membrane potential following a
presynaptic spike
 η: The change in the membrane potential following a postsynaptic
spike

f

(t  t j )  R  es /  (t  t jf  s)ds (3.14)
 Synaptic input at synpase
0
 The reset RI res   (t  t f ) (3.15)  (t  t f )  e(t t ) / (3.16) u(t f )   (3.17)
f
j
f
m
f
m
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3.2 The spike-response model (2)
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3.3 Spike time variability
Fig. 3.4 Normalized histogram of interspike
intervals (ISIs). (A) data from recordings of one
cortical cell (Brodmann’s area 46) that fired
without task-relevant characteristics with an
average firing rate of about 15 spikes/s. The
coefficient of variation of the spike trains is Cv ≈
1.09. (B) Simulated data from a Poisson
distributed spike trains I which a Gaussian
refractory time has been included. The solid line
represents the probability density function of the
exponential distribution when scaled to fit the
normalized histogram of the spike train. Note hat
the discrepancy for small interspike intervals is
due to the inclusion of a refractory time.




Neurons in brain do not fire regularly but seem extremely noisy.
Neurons that are relatively inactive emit spikes with low frequencies that
are very irregular.
High-frequency responses to relevant stimuli are often not very regular.
The coefficient of variation, Cv=σ/μ (3.18)
 Cv≈0.5-1 for regularly spiking neurons in V1 and MT
 Spike trains are often well approximated by Poisson process, Cv=1
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3.3.1 Biological irregularities

Biological networks do not have the regularities of the engineeringlike designs of the IF-neurons
 Consider irregularities from different sources in the biological
nervous system
 The external input to the neuron
 Structural irregularities

Use a statistical approach
3.3.2 Stochastic modeling


Noise can be described as a random variable
Use the probability density function (pdf) (see Appendix B).
 Normal distribution
 Poisson process



Mean
Variance
Higher moments of the distribution
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3.3.3 Normal distribution

Many random processes observed in nature are
 Gaussian bell curve


Normal distribution
N ( , )
Gaussian distribution
pdf normal ( x;  ,  )( x) 
1 ( x   ) 2 / 2 2
e
2
(3.19)
 Mean,
 Variance, 
 distribution
 Standard normal
or white noise,   0

The central limit theorem
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Fig. 3.5 A normalized histogram of 1000
random numbers and the functional form of
the corresponding probability distribution
functions (pdfs). (A) Random variables from
a normal distribution (Gaussian distribution
with mean μ = 0 and variace σ = 1). The solid
line represents the corresponding pdf (eqn
3.19). (B) Exponential distribution with
mean b = 2 (eqn 3.20)
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3.3.4 Poisson process

Exponential distribution
pdf exponential ( x;  )  ex (3.20)

Poisson distribution
Fig. 3.5
 The number of events when the time between events is
exponentially distributed
e
( x;  )   
i!
i 1
x
pdf

Poisson
i
(3.21)
A Poisson process
 Generating spike trains
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Fig. 3.5 A normalized histogram of 1000
random numbers and the functional form of
the corresponding probability distribution
functions (pdfs). (B) Exponential distribution
with mean b = 2 (eqn 3.20)
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3.4 Noise models for IF-neurons

Noise in the neuron models
 Stochastic threshold
     (1) (t ) (3.22)
 Random reset
u res  u res   ( 2) (t ) (3.23)
 Noisy integration
m

du
 u  RI ext   (3) (t )
(3.24)
dt
The stochastic process of a neuron
 Appropriate choices for the random
variables η(1), η(2), and η(3).
Fig. 3.6 Three different noise models of I&F neurons
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3.4.1 Simulating variabilities of real neurons

The appropriate choice of the random process, probability
distribution, time scale
 Cannot give general anwers
 Fit experimental data

Noise in IF model by noisy input.
I ext  I ext   with   N (0,1) (3.25)
 Central limit theorem

Lognormal distribution
pdf lognormal ( x;  ,  ) 
1
x 2
 (log(x )   ) 2
e
2 2
(3.26)
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Fig. 3.7 Simulated interspike interval (ISI) distribution of a leaky
IF-neuron with the threshold 10 and time constant τm=10. The
underlying spike train was generated with noisy input around the
mean value RI = 12. The fluctuation were therefore distributed
with a standard normal distribution. The resulting ISI histogram is
well approximated by a lognormal distribution (solid line). The
coefficient of variation of the simulated spike train is Cv ≈ 0.43
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3.4.2 Input spike trains

Simulation of an IF-neuron that has no internal noise but is
driven by 500 independent incoming Poisson spike trains.
EPSP amplitude
w=0.5
Firing
threshold
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w=0.25
Fig. 3.8 Simulation of IF-neuron
that has no internal noise but is
driven
by
500
independent
incoming spike trains with a
corrected Poisson distribution. (A)
The sums of the EPSPs, simulated
by an α-function for each incoming
spike with amplitude w = 0.5 for the
upper curve and w = 0.25 for the
lower curve. The firing threshold for
the neuron is indicated by the
dashed line. The ISI histograms
from the corresponding simulations
are plotted in (B) for the neuron
with EPSP amplitude of w = 0.5 and
in (C) for the neuron with EPSP
amplitude of w = 0.25.
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3.4.3 The gain function depends on input

The activation function of the neuron depends on the
variations in the input spike train.
 The average firing rate for a stochastic IF-neuron [Tuckewell, 1988]
r  (t ref   m 
(  R I ext ) / 
( u res  R I ext ) / 
 eu [1  erf (u )du ) 1
2
r  r (  ,  ,...) (3.28)
(3.27)
mean :   R I
variance : 
low σ: sharp transition
high σ: linearized
Fig. 3.9 The gain function of an IFneuron that is driven by an external
current that is given a normal
distribution with mean μ=RI and
variance σ. The reset potential was
set to Ures = 5 and the firing
threshold of the IF-neuron was set
to 10. The three curves correspond
to three different variance
parameters σ.
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Conclusion

Simplified neuron models
 Designed for the study of information processing in networks of
neurons.
 The information transmitted only by the occurrence of a spike.
 Integrate-and-fire neuron models
 A subthreshold leaky-integrator dynamic
 A firing threshold
 A reset mechanism

The variability in the firing times
 Noise models
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