Integrate and fire model
The integrate-and-fire model is the simplest model of a spiking neuron that takes into account the dynamics of the input. The basis of the integrate-and-fire model is the simple compartmental model of a neuron:
where Iin is the input current (e.g., from a synapse), C is the membrane capacitance, R is the net membrane resistance due to all passive channels (), and is the resting potential of the neuron (typically about -70 mV). Thus, if no current is injected (Iin=0) and the system has come to equilibrium then the membrane potential, , will equal the resting potential, .
Solving for the currents at the top-left node of this circuit we obtain
Grouping the terms involving on the left-hand side and multiplying both sides by R we obtain the following differential equation governing the relation between Iin and :
The left-hand side of equation 1 is just our familiar leaky-integrator. Its impulse response function is . Thus, we could compute the membrane voltage in response to the time-varying current by convolving with the right-hand side of equation 1. Since the first term () is just a constant and integrates to one, we obtain
where denotes convolution. Alternatively, we could compute the membrane voltage by simulating the differential equation directly in discrete-time (see handout entitled Simulating differential equations).
So far, everything about our model is passive and linear. We can make our simple neuron model spike by setting the membrane voltage equal to a large value, , (typically around +50 mV) once it exceeds a certain threshold (typically about -40 mV):
Then, immediately after a spike is emmited, the membrane voltage is set to :
where is the time-step of the simulation. Once the membrane voltage is reset to , we continue merrily along our way simulating the differential equation (1) until hits threshold again. Note however that we must simulate the differential equation directly in this case (i.e., not with convolution), because we do not want the spike to get filtered into future values of .