captured by greatly reduced approximations. Amongst other things, first pass approximations ignore the voltage-dependent nature of Tm and make further simplifications, but are still able to capture many of the important dynamics, such as neural depolarization. We now describe these, following the basic approach of Izhikevich (2005).

Morris-Lecar and Related 'Planar' Simplifications

An essential ingredient of a neural firing is a fast depolarizing current such as Na+ - which is turned on subsequent to a synaptic current - and a slow repolarizing current such as K+ - which restores the resting membrane potential. These in turn are facilitated by the existence of slow and fast ion channels of the respective species, Tm(V) << Tn(V). The depolarizing current represents positive feedback (i.e. is self promoting) and, if a threshold is reached before a sufficient number of slower K+ channels are open, the cell depolarizes. By contrast, the Na+ inactivation channel plays less of a "brute force" role and can be ignored. The requirement of a "fast" depolarizing current and a slow repolarizing current can be met in a two dimensional ("planar") system, dV

^ = QNam^ (V) x (V - VNa) + gKn (V) x (V - Vk) + ql x (V - VL) + I,

where the dynamics of the slow repolarizing K+ is given by dn (nTO — n) dt Tn and the steady state currents given by, n™(V) = 1 , nTXV, and m^(V) =

In other words, fast sodium channels instantaneously assume their steady state values following a change in membrane potential, hence adapting in a step-wise manner to a step-like change in membrane potential. Hence there is no differential equation for the Na+ activation channels, m. This is exactly the form of the Morris-Lecar model, with the exception of a substitution of Na+ currents with Ca++.

The system (35)-(36) is known as planar, as its phase space is the two-dimensional plane spanned by V (the abscissa) and n (the ordinate). To understand the dynamics we calculate the nullclines for the dynamical variables V and n. The V-nullcline, obtained by substituting dV/dt = 0 into (35) is,

Similarly, the n-nullcline, obtained by setting dn/dt = 0 is, nmax Cool n =1 + eVn-V)/a , (38)

These nullclines for the parameter values given in Table 1 and synap-tic current I = 0 are plotted in Fig. 15. We see that there are three null-clines crossings corresponding to three fixed points, { — 66,0}, { — 56,0} and { — 25,0.5}. Stability analysis shows that these fixed points are a stable focus, saddle point and spiral outset respectively (Izhikevich 2005). Hence the first fixed point first represents the only stable (steady state) solution.

Figure 15(b) shows two heteroclines - that is, outsets of the saddle point that become insets of the stable node. A long heterocline (magenta) traverses the nullclines before reaching the node whereas the shorter one (yellow) is able to track in parallel to the n-nullcline directly between the fixed points.

Table 1. Parameter values for the planar system (37)-(38) and figures 15-19

Capacitance,

Synaptic current (default), Leaky channels: Sodium channels:

Potassium channels:

"high threshold" VK = "low threshold" VK =

-90; gK = 10; K = -25; an --78; gK = 10; K = -45; an

Fig. 15. Fixed points and nullclines of the planar system (37)-(38). The V-nullcline (39) is given in blue and the n-nullcline (eqn 40) in black. (a) Fixed points occur at the intersections of the nullclines: Stable node (red), saddle point (blue) and spiral outset (yellow). Arrows show representative vector field. (b) Long (magneta) and short (yellow) heteroclines of stable node and saddle

In Fig. 16 is shown representative orbits of this system. Three "subthreshold" (green) and three "suprathreshold" (red) orbits are shown. In the latter case, the neuron depolarizes before returning to its resting state. It should be noted that this threshold depends not only on the initial membrane potential V but also the initial K+ membrane conductance. The separatrix between sub- and supra-threshold is constituted by the inset of the saddle point (not shown).

Whether the initial condition is sub- or supra-threshold, this system only has a single steady state solution in the current parameter regime. Hence, after at most one depolarization, it enters a quiescent state. Thereafter a discrete synaptic input, such as due to an excitatory post-synaptic potential (EPSP), will trigger a further discharge only if it is of sufficient strength to 'knock' the system over the inset of the saddle point. This will hence determine whether the resulting neural response is of the green or red waveform as in Fig. 16.

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