Hindmarsh & Rose mathematical model of neuronal bursting

The now classical Hindmarsh & Rose mathematical model of neuronal burstingwas first published in peer-reviewed form back in 1984, by ascientific periodical named “Proceedings of The Royal Society ofLondon. Series B, Biological Sciences”. (This seminal paper is nowonly available online via a JSTOR’s(probably institutional) subscription, so go check out with youruniversity. Alternatively, go to your university’s medical schoollibrary periodicals section.)

Hindmarsh & Rose work were initiated by the discovery of a neuronal cell in the brain of the pond snail Lymnae,which was initially “silent” (the molluscan burst neuron had beenpreviously hyperpolarized to stop the bursting), but when depolarizedby a short current pulse generated a burst that greatly outlasted thestimulus – i.e., an action potential followed by a slow depolarizingafter-potential.

In seeking an explanation for the phenomena also observed incrustaceans and vertebrates, the research collaborators devised asystem of 3 coupled (3-variable) first-order differential equationsof the following form:

dx/dt = y – f(x) – z – I,
dy/dt = g(x) – y,
dz/dt = r(h1(x) – z),

where

(the first 3 variables listed below are coordinates which representthe states of the dynamical system – a single neuron in this case -varying over time)

x: (neuron) membrane potential,
y: potential of the ionic channels subserving accomodation,
z: the slow adaptation current which moves the voltage in andout of the inherent bistable regime and which terminates spikedischarges,
r: the time scale of the slow adaptation current,
I: the applied current,
h1(x): the scale of the influence of the slow dynamics onmembrane potential, which determines whether the neuron fires in atonic or in a burst mode (when exposed to a sustained current input),
f(x): a cubic function,
g(x): not a linear function.

Depending on the values of the above parameters, neurons can be in asteady-state, they can generate a periodic low-frequency repetitivefiring, chaotic bursts, or high frequency discharges of actionpotentials. Despite its inherent single cell description, Hindmarsh& Rose neurons can be linked by introducing equations accountingfor electrical and/or chemical junctions which underlie syncronizationin experiment material on the cooperative behavior of neurons thatarises when cells belonging to large assemblies are coupled with eachother. Depending on the degree of coupling between the neurons, such alinkage can lead to out/in phase bursting or to a chaotic behavior.

H-r

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