where . If , then . In this situation the time scale that governs the evolution of is much faster than that of . This observation can be exploited for the analysis of the system. The general idea is that of a ‘separation of time scales’; in the mathematical literature the limit of is called ‘singular perturbation’. Oscillatory behavior for small is called a ‘relaxation oscillation’.
What are the consequences of the large difference of time scales for the phase portrait of the system? Recall that the flow is in direction of . In the limit of , all arrows in the flow field are therefore horizontal, except those in the neighborhood of the -nullcline. On the -nullcline, and arrows are vertical as usual. Their length, however, is only of order . Intuitively speaking, the horizontal arrows rapidly push the trajectory toward the -nullcline. Only close to the -nullcline can directions of movement other than horizontal are possible. Therefore, trajectories slowly follow the -nullcline, except at the knees of the nullcline where they jump to a different branch.
Excitability can now be discussed with the help of Fig. 4.21. A current pulse shifts the state of the system horizontally away from the stable fixed point. If the current pulse is small, the system returns immediately (i.e., on the fast time scale) to the stable fixed point. If the current pulse is large enough so as to put the system beyond the middle branch of the -nullcline, then the trajectory is pushed toward the right branch of the nullcline. The trajectory follows the -nullcline slowly upward until it jumps back (on the fast time scale) to the left branch of the -nullcline. The ‘jump’ between the branches of the nullcline corresponds to a rapid voltage change. In terms of neuronal modeling, the jump from the right to the left branch corresponds to the downstroke of the action potential. The middle branch of the -nullcline (where ) acts as a threshold for spike initiation; cf. Fig. 4.22.
If we are not interested in the shape of an action potential, but only in the process of spike initiation, we can exploit the separation of time scales for a further reduction of the two-dimensional system of equations to a single variable. Without input, the neuron is at rest with variables . An input current acts on the voltage dynamics, but has no direct influence on the variable . Moreover, in the limit of , the influence of the voltage on the -variable via Eq. (4.34) is negligible. Hence, we can set and summarize the voltage dynamics of spike initiation by a single equation
In a two-dimensional neuron model with separation of time scales, the upswing of the spike corresponds to a rapid horizontal movement of the trajectory in the phase plane. The upswing is therefore correctly reproduced by Eq. (4.35). The recovery variable departs from its resting value only during the return of the system to rest, after the voltage has (nearly) reached its maximum (Fig. 4.22A). In the one-dimensional system, the downswing of the action potential is replaced by a simple reset of the voltage variable, as we will see in the next chapter.
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