Subsections

• 3.3.1 Type I models
  • 3.3.1.1 Example: Canonical type I model
  
  
• 3.3.2 Type II models
• 3.3.3 Separation of time scales
  • 3.3.3.1 Example: Piecewise linear nullclines I
  • 3.3.3.2 Trajectory during a pulse (*)
  • 3.3.3.3 Example: Piecewise linear nullcline II

3.3 Threshold and excitability

We have seen in the previous chapter the Hodgkin-Huxley model does not have a clear-cut firing threshold. Nevertheless, there is a critical regime where the sensitivity to input current pulses is so high that it can be fairly well approximated by a threshold. For weak stimuli, the voltage trace returns more or less directly to the resting potentials. For stronger stimuli it makes a large detour, that is, emits a spike; see Fig. 3.10B. This property is characteristic for a large class of systems collectively termed excitable systems.

For two-dimensional models, excitability can be discussed in phase space in a transparent manner. We will pose the following questions. What are the conditions for a threshold behavior? If there is no sharp threshold, what are the conditions for a regime of high (threshold-like) sensitivity? We will see that type I models indeed have a threshold whereas type II models have not. On the other hand, even type II models can show threshold-like behavior if the dynamics of w is considerably slower than that of u.

Throughout this section we use the following stimulation paradigm. We assume that the neuron is at rest (or in a known state) and apply a short current pulse I(t) = q $\delta$(t) of amplitude q > 0. The input pulse influences the neuronal dynamics via Eq. (3.2). As a consequence, the voltage u jumps at t = 0 by an amount $\Delta$u = q R/$\tau$; the time course of the recovery variable w, on the other hand, is continuous. In the phase plane, the current pulse therefore shifts the state (u, w) of the system horizontally to a new value (u + $\Delta$u, w). How does the system return to equilibrium? How does the behavior depend on the amplitude q of the current pulse?

We will see that the behavior can depend on q in two qualitatively distinct ways. In type I models, the response to the input shows an `all-or-nothing' behavior and consists either of a significant pulse (that is, an action potential) or a simple decay back to rest. In this sense, type I models exhibit a threshold behavior. If the action potential occurs, it has always roughly the same amplitude, but occurs at different delays depending on the strength q of the stimulating current pulse. In type II models, on the other hand, the amplitude of the response depends continuously on the amplitude q. Therefore, type II models do not have a sharp threshold.

Note that even in a model with threshold, a first input pulse that lifts the state of the system above the threshold can be counterbalanced by a second negative input which pulls the state of the system back. Thus, even in models with a threshold, the threshold is only `seen' for the specific input scenario considered here, viz., one isolated short current pulse.

3.3.1 Type I models

Figure 3.9: Threshold in a type I model. A. The stable manifold (fat solid line) of the saddle point [open circle at about (u, w) = (- 0.4, - 0.3)] acts as a threshold. Trajectories (thin solid lines) that start to the right of the stable manifold, cannot return directly to the stable fixed point (filled circle) but have to take a detour around the repulsive fixed point [open circle at (u, w) = (0.7, 0.6)]. The result is a spike-like excursion of the u-variable. Thin dashed lines are the nullclines; the fat dashed line is the unstable manifold of the saddle point. B. Blow-up of the rectangular region in A. The starting points of the two sample trajectories are marked by small dots.

As discussed above, type I models are characterized by a set of three fixed points, a stable one to the right, a saddle point in the middle, and an unstable one to the left. The linear stability analysis at the saddle point reveals, by definition of a saddle, one positive and one negative eigenvalue, $\lambda_{+}^{}$ and $\lambda_{-}^{}$, respectively. The imaginary part of the eigenvalues vanishes. Associated with $\lambda_{-}^{}$ is the (real) eigenvector $\vec{{e}}_{-}^{}$. A trajectory which approaches the saddle in direction of $\vec{{e}}_{-}^{}$ from either side will eventually converge towards the fixed point. There are two of these trajectories. The first one starts at infinity and approaches the saddle from below. The second one starts at the unstable fixed point and approaches the saddle from above. The two together define the stable manifold of the fixed point (Verhulst, 1996; Hale and Koçak, 1991). A perturbation around the fixed point that lies on the stable manifold returns to the fixed point. All other perturbations will grow exponentially.

The stable manifold plays an important role for the excitability of the system. Due to the uniqueness of solutions of differential equations, trajectories cannot cross. This implies that all trajectories with initial conditions to the right of the stable manifold must make a detour around the unstable fixed point before they can reach the stable fixed point. Trajectories with initial conditions to the left of the stable manifold return immediately towards the stable fixed point; cf. Fig. 3.9.

Let us now apply these considerations to models of neurons. At rest, the neuron model is at the stable fixed point. A short input current pulse moves the state of the system to the right. If the current pulse is small, the new state of the system is to the left of the stable manifold. Hence the membrane potential u decays back to rest. If the current pulse is sufficiently strong, it will shift the state of the system to the right of the stable manifold. Since the resting point is the only stable fixed point, the neuron model will eventually return to the resting potential. To do so, it has, however, to take a large detour which is seen as a pulse in the voltage variable u. The stable manifold thus acts as a threshold for spike initiation.

3.3.1.1 Example: Canonical type I model

For I < 0 on the right-hand side of Eq. (3.30), the phase equation d$\phi$/dt has two fixed points. The resting state is at the stable fixed point $\phi$ = $\phi_{r}^{}$. The unstable fixed point at $\phi$ = $\vartheta$ acts as a threshold; cf. Fig. 3.8.

Let us now assume initial conditions slightly above threshold, viz., $\phi_{0}^{}$ = $\vartheta$ + $\delta$$\phi$. Since d$\phi$/dt|_{$\phi_{0}$} > 0 the system starts to fire an action potential but for $\delta$$\phi$ $\ll$ 1 the phase velocity is still close to zero and the maximum of the spike (corresponding to $\phi$ = $\pi$) is reached only after a long delay. This delay depends critically on the initial condition.

3.3.2 Type II models

In contrast to type I models, Type II models do not have a stable manifold and, hence, there is no `forbidden line' that acts as a sharp threshold. Instead of the typical all-or-nothing behavior of type I models there is a continuum of trajectories; see Fig. 3.10A.

Nevertheless, if the time scale of the u dynamics is much faster than that of the w-dynamics, then there is a critical regime where the sensitivity to the amplitude of the input current pulse can be extremely high. If the amplitude of the input pulse is increased by a tiny amount, the amplitude of the response increases a lot (`soft' threshold).

Figure 3.10: Threshold behavior in a type II model. A. Trajectories in the phase starting with initial conditions (u₀, w_rest) where u₀ = - 0.5, - 0.25, - 0.125, 0, 0.25. B. Projection of the trajectories on the voltage axis. For u₀$\le$ - 0.25, the trajectories return rapidly to rest. The trajectories with u₀$\ge$ - 0.1 develop a voltage pulse. Parameters as in Fig. 3.5 with I = 0.

In practice, the consequences of a sharp and a `soft' threshold are similar. There is, however, a subtle difference in the timing of the response between type I and type II models. In type II models, the peak of the response is always reached with roughly the same delay, independently of the size of the input pulse. It is the amplitude of the response that increases rapidly but continuously; see Fig. 3.10B. On the other hand, in type I model the amplitude of the response is rather stereotyped: either there is an action potential or not. For input currents which are just above threshold, the action potential occurs, however, with an extremely long delay. The long delay is due to the fact that the trajectory starts in the region where the two fixed points (saddle and node) have just disappeared, i.e., in a region where the velocity in phase space is very low.

3.3.3 Separation of time scales

Consider the generic two-dimensional neuron model given by Eqs. (3.2) and (3.3). We measure time in units of $\tau$ and take R = 1. Equations (3.2) and (3.3) are then

$${{\text{d}}u\over {\text{d}}t}$$ = F(u, w) + I (3.31)

$${{\text{d}}w \over {\text{d}}t} \epsilon$$ = (3.32)

where $\epsilon$ = $\tau$/$\tau_{w}^{}$. If $\tau_{w}^{}$ $\gg$ $\tau$, then $\epsilon$ $\ll$ 1. In this situation the time scale that governs the evolution of u is much faster than that of w. 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 $\epsilon$$\to$ 0 is called `singular perturbation'. Oscillatory behavior for small $\epsilon$ 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 ($\dot{{u}}$,$\dot{{w}}$). In the limit of $\epsilon$$\to$ 0, all arrows in the flow field are therefore horizontal, except those in the neighborhood of the u-nullcline. On the u-nullcline, $\dot{{u}}$ = 0 and arrows are vertical as usual. Their length, however, is only of order $\epsilon$. Intuitively speaking, the horizontal arrows rapidly push the trajectory towards the u-nullcline. Only close to the u-nullcline directions of movement other than horizontal are possible. Therefore, trajectories slowly follow the u-nullcline, except at the knees of the nullcline where they jump to a different branch.

Figure 3.11: Excitability in a type II model with separated time scales. The u-dynamics are much faster than the w-dynamics. The flux is therefore close to horizontal, except in the neighborhood of the u-nullcline (schematic figure). Initial conditions (circle) to the left of the middle branch of the u-nullcline return directly to the stable fixed point; a trajectory starting to the right of the middle branch develops a voltage pulse.

Excitability can now be discussed with the help of Fig. 3.11. 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 u-nullcline, then the trajectory is pushed towards the right branch of the u nullcline. The trajectory follows the u-nullcline slowly upwards until it jumps back (on the fast time scale) to the left branch of the u-nullcline. The `jump' between the branches of the nullcline correspond to a rapid voltage change. In terms of neuronal modeling, the jump from the right to the left branch corresponds to downstroke of the action potential. The middle branch of the u nullcline (where $\dot{{u}}$ > 0) acts as a threshold for spike initiation. This is shown in the simulation of the FitzHugh-Nagumo model in Fig. 3.12.

Figure 3.12: FitzHugh-Nagumo model with separated time scales. All parameters are identical to those of Fig. 3.10 except for $\epsilon$ which has been reduced by a factor of 10. A. A trajectory which starts to the left-hand side of the middle branch of the u-nullcline, returns directly to the rest state; all other trajectories develop a pulse. B. Due to slow w dynamics pulses are much broader than in Fig. 3.10.

3.3.3.1 Example: Piecewise linear nullclines I

Let us study the piecewise linear model shown in Fig. 3.13,

$${{\text{d}}u\over {\text{d}}t}$$ = f (u) - w + I (3.33)

$${{\text{d}}w \over {\text{d}}t} \epsilon$$ = (3.34)

with f (u) = a u for u < 0.5, f (u) = a (1 - u) for 0.5 < u < 1.5 and f (u) = c₀ + c₁ u for u > 1.5 where a, c₁ < 0 are parameters and c₀ = - 0.5a - 1.5c₁. Furthermore, b > 0 and 0 < $\epsilon$ $\ll$ 1.

The rest state is at u = w = 0. Suppose that the system is stimulated by a short current pulse that shifts the state of the system horizontally. As long as u < 1, we have f (u) < 0. According to (3.33), $\dot{{u}}$ < 0 and u returns to the rest state. For u < 0.5 the relaxation to rest is exponential with u(t) = exp(a t) in the limit of $\epsilon$$\to$ 0. Thus, the return to rest after a small perturbation is governed by the fast time scale.

If the current pulse moves u to a value larger than unity, we have $\dot{{u}}$ = f (u) > 0. Hence the voltage u increases and a pulse is emitted. That is to say, u = 1 acts as a threshold.

Let us now assume that an input spike from a presynaptic neuron j arrives at time t_j^(f). Spike reception at the neuron generates a sub-threshold current pulse I(t) = c $\delta$(t - t_j^(f)), where 0 < c $\ll$ 1 is the amplitude of the pulse. The perturbation causes a voltage response,

$$\kappa \left(\vphantom{ - {t-t_j^{(f)}\over \tau_a} }\right. {t-t_j^{(f)}\over \tau_a} \left.\vphantom{ - {t-t_j^{(f)}\over \tau_a} }\right)$$ (3.35)

with $\tau_{a}^{}$ = - 1/a. If several input pulses arrive in a interval shorter than $\tau_{a}^{}$, then the responses are summed up and move the neuronal state beyond the middle branch of the u-nullcline. At this point a spike is triggered.

Figure 3.13: Piecewise linear model. The inset shows the trajectory (arrows) which follows the u nullcline at a distance of order $\epsilon$.

3.3.3.2 Trajectory during a pulse (*)

We have argued above that, during the pulse, the trajectory is always pushed towards the u-nullcline. We will show in this paragraph that the trajectory (u(t), w(t)) of the piecewise linear system (3.33)-(3.34) follows the u-nullcline w = f (u) + I at a distance of order $\epsilon$. We set

$$\epsilon \mathcal {O} \epsilon^{2}_{}$$ (3.36)

where x(u) is the momentary distance at location u. We show that (3.36) gives a consistent solution.

To do so we use Eq. (3.36) on the right-hand-side of (3.33) and (3.34). Thus, along the trajectory

$${{\text{d}}u\over {\text{d}}t} \epsilon$$ = (3.37)

$${{\text{d}}w \over {\text{d}}t} \epsilon$$ = (3.38)

where we have neglected terms of order $\epsilon^{2}_{}$. On the other hand, the derivative of (3.36) is

$${{\text{d}}w \over {\text{d}}t} {df\over du} {{\text{d}}u\over {\text{d}}t} \epsilon {{\text{d}}x\over {\text{d}}u} {{\text{d}}u\over {\text{d}}t}$$ (3.39)

We solve (3.38) and (3.39) for du/dt. The result to order $\epsilon$ is

$${{\text{d}}u\over {\text{d}}t} {\epsilon \over {df/ du}} \left[\vphantom{ b\,u - f(u) - I}\right. \left.\vphantom{ b\,u - f(u) - I}\right]$$ (3.40)

Comparison with (3.37) shows that the distance x is indeed of order one.

3.3.3.3 Example: Piecewise linear nullcline II

Let us study the relaxation to the stable fixed point after a pulse in the piecewise linear model. We use f' = a for the slope of the u-nullcline and b for the slope of the w-nullcline. Evaluation of (3.40) gives

$${{\text{d}}u\over {\text{d}}t} \epsilon \left(\vphantom{1-{b\over a} }\right. {b\over a} \left.\vphantom{1-{b\over a} }\right)$$ (3.41)

The decay is exponential with a time constant of recovery $\epsilon^{{-1}}_{}$ [1 - (b/a)]^-1. Hence the relaxation to the resting potential is governed by the slow recovery dynamics with a time scale of order $\epsilon$. The slow relaxation is one of the causes of neuronal refractoriness.

Similarly, the voltage during the spike is given by integrating

$${{\text{d}}u\over {\text{d}}t} \epsilon \left(\vphantom{1-{b\over c_1}}\right. {b\over c_1} \left.\vphantom{1-{b\over c_1}}\right) \epsilon {c_0 + I\over c_1}$$ (3.42)

Let us denote by $\hat{{t}}$ the time when the spike was triggered (i.e., when u crossed the middle branch of the u-nullcline). After the voltage increase during the initial phase of the limit cycle, the state of the system is on the right branch of the u-nullcline. There it evolves according to (3.42). When it reaches the knee of the nullcline, it jumps to the left branch where it arrives at time t_left. On the left branch, the relaxation to the resting potential is governed by (3.41). If we neglect the time needed for the jumps, the voltage during the limit cycle is therefore

$$\eta \hat{{t}} \left\{\vphantom{ \begin{array}{ccc} (u_{\rm right}-\overline{u})... ...\tau_{\rm recov}}\right) + u_r &{\rm for}& t_{\rm left} < t \end{array}}\right. \begin{array}{ccc} (u_{\rm right}-\overline{u}) \, \exp\left( - {... ...t}\over \tau_{\rm recov}}\right) + u_r &{\rm for}& t_{\rm left} < t \end{array}$$ (3.43)

with $\tau_{{\rm spike}}^{}$ = $\epsilon^{{-1}}_{}$ [1 - (b/c₁)]^-1, $\tau_{{\rm recov}}^{}$ = $\epsilon^{{-1}}_{}$ [1 - (b/a)]^-1 and parameters u_right = 1.5 + a/c₁, u_left = - 0.5, $\overline{{u}}$ = (c₀ + I)/(b - c₁), and u_r = I/(b - a). The representation of neuronal dynamics in terms of a response function as in equation (3.35) and a recovery function as in (3.43) plays a key role in formal spiking neuron models discussed in the following chapter.