Integral equation of neurons with absolute refractory period .
(i) Apply the population equation ( 14.5 ) to SRM neurons with escape noise which have an absolute refractory period
(ii) Introduce the normalization condition Eq. ( 14.8 ) so as to arrive at the Wilson-Cowan integral equation ( 14.10 ).
(iii) Use Eq. ( 14.8 ) to show that the mean interspike interval of neurons firing stochastically with a rate
where is a constant input potential.
Gain function of SRM neurons
Consider SRM neurons with escape noise such that the hazard function is given by with .
(i) Show that the survivor function in the asynchronous state is
(ii) Using your results in (i), find the gain function for neurons.
Hint: Remember that the mean firing rate for fixed is the inverse of the mean interval. You will have to use the lower incomplete gamma function .
Linearization of the Wilson Cowan integral equation .
The aim is to find the frequency dependent gain for a population of neurons with absolute refractoriness.
(i) Start from the Wilson-Cowan integral equation and linearize around a stationary state .
(ii) Start with the filter in Eq. ( 14.60 ) and derive directly the filter .
Slow noise in the parameters.
Consider a population of leaky integrate-and-fire neurons with time constant and resistance , driven by a constant super-threshold input . After each firing, the membrane potential is reset to which is chosen randomly from a distribution with mean .
(i) Calculate the interspike interval for a neuron which was reset at time to a value and that of another neuron which was reset at to .
(ii) Suppose a Gaussian distribution of reset values with standard deviation . Show that the standard deviation of the interval distribution is where is the derivate of the membrane potential at the moment of threshold crossing.
Linear response filter with step-function escape rate .
Consider , i.e., a step-function escape rate. For neurons fire immediately as soon as and we are back to a noise-free sharp threshold. For finite , neurons respond stochastically with time constant . We will show that
The neuron mode is a SRM with arbitrary refractoriness driven by a constant input and a time-dependent component . The total membrane potential at time is where .
(i) Show that the kernel for neurons with step-function escape rate is an exponential function.
[Hint: Denote by the time between the last firing time and the formal threshold crossing, The derivative of is a -function in time. Use a short-hand notation and exploit Eq. ( 14.58 ). ]
(ii) Calculate the liner filter and the response to an input current .
© Cambridge University Press. This book is in copyright. No reproduction of any part of it may take place without the written permission of Cambridge University Press.