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the input is sufficiently large. The frequency of any such chain of discharges increases with the magnitude of the synaptic input. On the other hand, the Hopf route will generate either damped, sub-threshold oscillations or a chain of depolarizations, although the frequency of these will be more-or-less constant. In the presence of discrete synaptic inputs, the saddle-node system will generate an all-or-nothing depolarization - or chain of depolarizations - if the input is sufficiently large. The frequency of any such chain of discharges increases with the magnitude of the synaptic input. On the other hand, the Hopf route will generate either damped, sub-threshold oscillations or a chain of depolarizations, although the frequency of these will be more-or-less constant. As discussed in Izhikevich (2005) these two distinct neuronal responses to applied (or synaptic) currents were first observed empirically by Hodgkin in the 1940's. Specifically, he classified neurons that showed a frequency-dependence on the size of the synaptic current (i.e. Hopf-like responses) as Type I neurons. In particular, for small currents, these neurons begin to fire at very slow frequencies. In contrast, those neurons that start firing at relatively rapid rates following a supra-threshold input - and which show very little further increases in frequency - were classified as Type II neurons. The squid axon described by the original Hodgkin-Huxley model (1952) is a representative example of a neuron with type II behavior.

The FitzHugh-Nagumo Model

As we have seen above, the shape and intersections of the nullclines plays the determining role in the behavior and bifurcations of the dynamics. In fact, all that is required to reproduce the qualitative nature of the dynamics is the cubic-like shape of the ^-nullcline and the presence of an n-nullcline with the appropriate intersections. Mathematically, these requirements can be met with the much simpler algebraic equations (FitzHugh 1961, Nagumo et al. 1962), dt = x (a - x)(x - 1) - y + I, = bx - cy, (39)

which have the simple nullclines, y = x (a — x)(x — 1) + I, y = b/cx, (40)

In Fig. 21 is illustrated a phase portrait and time series for this system following a super-critical Hopf bifurcation of the single fixed point. This system -and variations of it - are known as the FitzHugh-Nagumo model.

This system hence allows a closed-form analysis, with relatively simple algebraic forms, of the same qualitative phenomena as the planar model of Hodgkin-Huxley dynamics.

The Hindmarsh-Rose Model

The Hindmarsh-Rose model is the last in the "microscopic" domain for consideration. It continues the logic of the FitzHugh Nagumo model - namely that it captures the qualitative essence of neuronal firing through a simple algebraic form of the evolution equations (and hence of the nullclines). However, rather than further reducing the Hodgkin-Huxley model, the Hindmarsh-Rose (1984) model introduces an extra property. The system is given by, dx 3 2 T

When r = 0, the third variable plays no role and the system reduces to a variation of a FitzHugh Nagumo model - that is, a two dimensional spiking neuron with a simple algebraic form: An example is given in Fig. 22.

However, setting r > 0 but small has the effect of introducing the third variable into the dynamics. Notice that z only enters into the first two

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