One step forward more simplified neuron modelling is the "Integrate and Fire" model. Here, a neuron is reduced to its most basic function: fire action potentials, or not. Most of the physiological and mathematical richness of the Hodgkin-Huxley model (and the FHN model as per Question 1, to some extent) is removed for the benefit of computational efficiency. This becomes important as the number of neurons in a simulation increases to match what is seen in neural tissue (about 105 per mm3!).

In its most basic form, the Integrate and Fire neuron obeys the following ODE

 

dVdt=−(V−Vreset)+I

 

(notice the similarity with exponential growth models!)

However, once the membrane potential V crosses some threshold value Vo, a spike is said to have occured, and the membrane potential is reset to some baseline value Vreset. For the purpose of this exercise the threshold for spiking is Vo=−30mV, and the reset potential if Vreset=−80mV

a) (2 point) A sequence of action potentials in time is called a "spike train" (call it X). During the simulation of the integrate and fire model, the timing of action potentials (i.e. when the membrane potential crosses the threshold) is "saved" : if a spike occured at time step t, the spike train value for that time is equal to 1 (i.e. X=1). If no spike occured for that time step, that value is zero (i.e. X=0). Using the Euler method and for an input value of I=70mV, integrate the system and plot the spike train X as a function of time for a duration of 10ms with time steps of dt=0.01. The initial condition is V(0)=V_reset.

 

b)(2 points) A way to characterize neural responses is through the so-called "frequency response" curve. Generally, the more input delivered to a cell, the more active it will becomes (i.e. it will fire more action potentials per unit time, hence the use of the word frequency). Using your numerical integration scheme developed in a), use a for loop and count the number of action potentials fired as a function of I. Plot the frequency response curve of neuron for inputs I ranging from 0 to 100mV by steps of 1mV.