Izhikevich Neuron Part 4: Bursting and Bistability
As we have mentioned repeatedly, the Izhikevich neuron is incredibly functionally diverse, and is capable of capturing a wide arrange of spiking phenomena. This includes bursting neurons, and bistable neurons. In order to capture these behaviors in the Izhikevich neuron we need to make use of the reset mechanism in the model neuron. Recall from part one of the series, the Izhikevich model has 4 key parameters, $a$, $b$, $c$, and $d$. In parts two and three we discussed in detail how the parameters $a$ and $b$ effected the dynamics near rest. However, we did not discuss the importance of the parameters $c$ and $d$. As it turns out, these parameters control the reset of the membrane potential $V$ and the gating variable $W$. By manipulating these two parameters, we can generate a diverse array of phenomena such as phasic bursting, tonic bursting and bistability (see figure 1). Interestingly, all three of these phenomena can be understood as the interplay between the separatrix, the gating variable $W$, and the parameters $c$, $d$.
 |
| Figure 1. Sample firing patterns for the Izhikevich neuron. From left to right: A bistable neuron has a downstate when the neuron is at rest, and a spiking upstate where the neuron is stably spiking. A current pulse is sufficient to switch the neuron from the rest state into the spiking state. Phasic Burst: A phasic burst is a cluster of a few spikes in response to a brief current pulse. After the burst, the neuron returns to rest. Tonic Burst: a tonic burst is a periodically recurring clusters of spikes followed by a quiescent state. |
First, let us define what exactly we mean by bursting, and bistability. Bursting is defined as a cluster of several spikes followed by a period of inactivity called quiescence (see figure 2). bursting can be split into two sub-types: phasic bursts and tonic bursts. Phasic bursts occur when a brief pulse of current causes a burst of spikes followed by a return to rest. Tonic bursts on the other hand, are periodic cycles between an active spiking state and a silent quiescent state. Bursting is ubiquitous across the brain, being integral in robust signalling in the presence of noise, generating stable periodic motion such as breathing, and bursts can make use of downstream resonating neurons (see part 3).
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| Figure 2. The schematic of a tonically spiking Izhikevich neuron. This has been recreated from figure 9.5 in Dr. Izhikevich's book. A burst is composed of two phases, an silent quiescent phase, and an active spiking phase. The interburst period referred to the periodicity of the quiescent and active phase. During the active phase the time between the spikes is called the intraspike period. |
Bistability is quite different than bursting. This is defined by the coexisting of two different basin's of attraction (see here). In a single neuron context, bistability is the coexistence of a stable equilibrium point and a periodic limit cycle. This is quite common in many neurons, that have spiking up-states and quiescent down-states.
Interestingly the Izhikevich model can unify all three of these different behaviors by considering the reset of the model. Recall the equations for the Izhikevich model is given by
$$
C_m \frac{dV}{dt} = k (V-V_r)(V-{th}) - W + I
$$
$$
\frac{dW}{dt} = a(b(V-V_r) -W)
$$
$$
\text{When } V>V_{spike}\text{, }V\rightarrow c\text{, } W \rightarrow W+d
$$
It is useful to think about where the solution returns to after the spike, relative to the separatrix. If you are unfamiliar with the separatrix and phase plane analysis, I strongly suggest reading part 1. As a quick reminder, the separatrix (dashed black line in figures 3-5) is a line dividing the initial conditions that return to rest (gray), and the initial conditions that lead to a spike(white). If we plot the separatrix, and the reset value (red dots), we can easily tell the difference between phasic bursting, and bistability. Take a look at bistability (fig 3); the reset values tend to accumulate below the separatrix (white region).
 |
| Fig 3. The phase plane of a bistable neuron. Blue is the nullcline for Voltage, yellow is the nullcline for the $W$-gate. The dashed black line is the separatrix between, the rest state (grey) and spiking state (white). The red dots represent the reset value for each spike. As time approaches infinity, the red dots approach an equilibrium. Thus, once the solution enters the white spiking region it stays there indefinitely. |
This is a fixed point of a discrete difference equation. Because the reset for voltage is $c$ we can consider a discrete system:
$$
W_{n+1} = F(W_n) + d
$$
where
$F(W_n)$ is the change in $W$ as the solution of the original system, with initial conditions $V(0)= c$ and $W(0) = W_n$. Because $V$ is only reset to $c$ the value of $F(W_n)$ only depends on the reset value of the last spike. When the system is bistable
$$
W_{n+1} = F(W_n) + d
$$
has only a single stable fixed point, and that fixed point is stable. We can see this quite clearly by noticing that after each spike, $W_{n+1}$ and $W_n$ get closer and closer together. This is called spike accommodation, and is common in the nervous system. Thus bistable systems exist whenever the reset value approaches a stable fixed point.
With this in mind we can now easily create a phasic burst. Specifically by increasing the value of $d$ such that the fixed point of
$$
W_{n+1} = F(W_n) + d
$$
can be pushed above the separatrix, where no fixed points exist. Thus the last spike in the burst will put the reset value above the separatrix (gray region). Therefore, the solution will return to rest. We can also accomplish this by moving $c$ as well, as a more negative $c$ requires a smaller $d$ to change a bistable system into a phasic burster.
 |
| Fig 4. The phase plane of a phasic bursting neuron. Blue is the nullcline for
Voltage, yellow is the nullcline for the $W$-gate. The dashed black line
is the separatrix between, the rest state (grey) and spiking state
(white). The red dots represent the reset value for each spike. After 5 spikes the last reset is above the separatrix and the neuron will return to rest. Therefore the spiking state is not stable, and the solution only transiently passes through it. |
It becomes pretty easy to manipulate this system into a tonic
burster. Notice how the phasic system has a stable equilibrium. This
corresponds to the rest state. However, by increasing the injected
current $I$ we can shift the quadratic bowl upwards, such that there are
no equilibrium points (this corresponds to a saddle-node bifurcation).
Now, all initial conditions lead to spiking solutions. However, we can
still divide the system into two domains (figure 5). The quiescence
phase is in grey, and is the region where the solution isn't actively
firing. The white region is the active zone, where the voltage is
rapidly increasing and corresponds to a spiking zone. Notice here, that
each spike increases the value in $W_{n+1}$ just like before. Moreover,
the last spike in burst causes the value to increase above the dashed
line. This means that the solution must decrease in voltage and loop
back around the quadratic minimum. However, because there is no
equilibrium point in the quiescent region, the system must still
eventually spike, thus creating a tonic burster.
 |
| Fig 5. The phase plane of a tonic bursting neuron. Blue is the nullcline for Voltage, yellow is the nullcline for the $W$-gate. The dashed black line is the separatrix between, the quiescent state (grey) and active state (white). Here we define the active state as any initial condition where the spike will occur within 30 milliseconds. The red dots represent the reset value for each spike. After 3 spikes the last reset is above the separatrix and the neuron will need to enter the quiescent state. Because of the lack of equilibria, the solution well eventually leave the quiescent state and enter the active state. |
This approach of visualizing phase planes is a powerful approach in computational neuroscience, and is especially useful in understanding single neuron dynamics. As we can see, understanding how a parameter, like $c$, $d$, and $I$ effects the phase plane's geometry allows us to easily construct systems with certain computational properties like, phasic bursting, tonic bursting and bistability.
On a final note, I will leave you with a question for next blog article. Given that we created a discrete system
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