Thalamocortical circuit

Information from the sensory surround transmitts through the nuclei of the thalamus en route to the cortex. The thalmic reticular nucleus sits between these structures and shapes thalamic activity through the inhibitory synapses sent to thalamus.

Electrical synapses

Electrical synapses are a major class of synapse formed by gap junctions between neurons. These gap junctions allow for electrical current to flow between two coupled neurons, allowing voltage differences to influence the activity of the cells.

A simple model of an electrical synapse can thus be a static resistance applied to the voltage difference between two neurons:

Computational models

My expertise with computational modelling is with utilizing the Hodgkin-Huxley (HH) based models. Neuron spiking is simulated from the biophysics of the sodium and potassium ion channels and their gating properties, mathematically described by Hodgkin and Huxley.

The simplest example is shown here:

A basic solving interface for this system could be written as:

using OrdinaryDiffEq

prob = ODEProblem(dsim!, u0, (startTime, endTime), p)
# dsim: HH function to be solved
# u0: initial conditions 
# p: parameters (channels, synapses etc.)

sol = solve(prob, BS3(), saveat=dt)

where parameters (p) are:

struct HHmodel
    gNa::Float64
    gK::Float64
    gL::Float64

    ENa::Float64
    EK::Float64
    EL::Float64

    C::Float64

    Iapp::Float64
end

and with the function dsim! describing the HH equations:

function dsim!(du, u, p, t)
	v, n, m, h = u

    I = (p.gK * (n^4.0) * (v-p.EK))
      + (p.gNa * (m^3.0) * h * (v-p.ENa))
      + (p.gL * (v-p.EL)) 
      + p.Iapp

    du[1]  = (-1.0/p.C) * I
    du[2]  = dn(v,n)
    du[3]  = dm(v,m)
    du[4]  = dh(v,h)
end
#dn,dm & dh dynamics ommited for clarity 

Solving the system with a positive square current illicits spiking such as:

Coming soon: An interactive parameter explorer for neuron models