The now classical Hindmarsh & Rose mathematical model of neuronal bursting was first published in peer-reviewed form back in 1984, by a scientific periodical named "Proceedings of The Royal Society of London. Series B, Biological Sciences". (This seminal paper is now only available online via a JSTOR's (probably institutional) subscription, so go check out with your university. Alternatively, go to your university's medical school library 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 been previously hyperpolarized to stop the bursting), but when depolarized by a short current pulse generated a burst that greatly outlasted the stimulus - i.e., an action potential followed by a slow depolarizing after-potential.

In seeking an explanation for the phenomena also observed in crustaceans and vertebrates, the research collaborators devised a system of 3 coupled (3-variable) first-order differential equations of 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 represent the 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 and out of the inherent bistable regime and which terminates spike discharges,

*r*: the time scale of the slow adaptation current,

*I*: the applied current,

*h1(x)*: the scale of the influence of the slow dynamics on membrane potential, which determines whether the neuron fires in a tonic 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 a steady-state, they can generate a periodic low-frequency repetitive firing, chaotic bursts, or high frequency discharges of action potentials. Despite its inherent single cell description, Hindmarsh & Rose neurons can be linked by introducing equations accounting for electrical and/or chemical junctions which underlie syncronization in experiment material on the cooperative behavior of neurons that arises when cells belonging to large assemblies are coupled with each other. Depending on the degree of coupling between the neurons, such a linkage can lead to out/in phase bursting or to a chaotic behavior.