08.Integrate and Fire Models Part 1

Integrate-and-fire Models

Integrate-and-fire Model

We call this integrate and fire because we assume that the presynaptic current will come in and superpose on each other and push the membrane potential to threshold, as then the neuron will fire.

This is explicitly demonstrated in the equations. We have a current as total current $$I$$.

Most simple integrate-and-fire model is a soma constructed from a resistance and capacitor,

$$ R C\frac{du}{dt} = - u + R I(t). $$

To generalize it we can introduce non-linear forms

$$ \frac{du}{dt} = F(u) + G(u)I(t). $$

Leaky Integrate-and-fire Model

Nonlinear Integrate-and-fire Model

The change of membrane potential doesn’t depend on the membrane potentials linearly anymore.

$$ \tau \frac{d u}{dt} = F(u) + G(u) I. $$

As an example, //quadratic// model by Latham et al, 2000, Feng, 2001, Hansel and Mato, 2001, use a quadratic dependence

$$ \tau \frac{du}{dt} = a_0 (u - u_{rest})(u - u_c) + I R. $$

Spike Response Mdoel

Just a nonlinear integrate-and-fire model.

$$ u_i(t) = \eta(t-\hat t_i) + \sum_j w_{ij} \sum_j \epsilon_{ij} (t-\hat t_i, t-\hat t_j^{(f)}) + \int_0^\infty \kappa(t-\hat t_i,s) I^{ext}(t-s) ds, $$

where $$\eta(t-\hat t_i)$$ is the evolution of action potential for a firing, while $$\epsilon_{ij} (t-\hat t_i, t-\hat t_j^{(f)})$$ is the response of neuron i with stimulation from neuron j, $$w_{ij}$$ is called efficacy.

There are many interesting properties about SRM model.

1. threshold depends on $$t-\hat t_i$$;

The biological interpretation of the //response kernels// ($$\eta$$, $$\epsilon$$, and $$\kappa$$) are explained in Gerstner’s book, section 4.2.1.

Here are the important questions to ask.

1. What are the interpretation of kernels?
2. How to model refractoriness using SRM? (In general, by tuning the three kernels)

FitzHugh-Nagumo model (refractoriness)

How exactly does hyperpolarization help us to understand refractoriness?

Relation to Leaky Integrate-and-fire Model

SRM0

Removing some of the possible refractoriness from kernels $$\kappa$$ and $$\epsilon$$.