**VS 298
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 *I _{in}* is the input current (e.g., from a synapse),

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 *I _{in}* and :

(1)

where .

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

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 .