Izhikevich Neurons, Part 5: Post-inhibitory rebound

 

Part 5: Post-Inhibitory Rebound

Post-Inhibitory Rebound (also called rebound excitation) is the flip side of the usual excitation story. Instead of firing because you pushed the neuron up, the neuron fires because you stopped pushing it down. Hold a cell hyperpolarized long enough, release it, and you get a spike, or a burst of spikes. It's a signal for the offset of inhibition rather than the onset of excitation, and it's a handy trick for a circuit: it lets a cell announce "the brakes just came off" without needing any excitatory input at all. Neurons in VTA and SNc do this, and it shows up all over thalamus too (called low threshold spikes in the 80s). Classically thalamocortical relay cells are the textbook case, where a burst of inhibition from reticular thalamic neurons is followed by a rebound burst that helps drive oscillations during sleep spindles.

Izhikevich Neurons are perhaps one of the simplest models of Post-inhibitory rebound. Lets see how to construct a rebound burster. 

Two phase planes

Post-inhibitory rebound is easiest to understand by separating the input into two phase planes: one at baseline current, one at baseline plus inhibition.

At baseline current, the neuron has a separatrix just like the one from Part 4: grey region where trajectories fall back to rest, white region where they fire. The stable equilibrium (black dot) sits right at the edge of that grey region. That V nullcline is very close to a bifurcation, barely grazing the W nullcline. Check the eigenvalues there (a complex conjugate pair, −0.09 ± 0.3i) and you'll find the neuron is a resonator (Part 3). That's what makes the separatrix nonmonotonic and pulls it in tight around the equilibrium, so the grey region barely wraps around the black dot before giving way to white. The complex pair isn't strictly required for rebound, but it's a reliable way to get an equilibrium that sits this close to its own threshold. The upshot: a point just slightly hyperpolarized relative to rest is already across the separatrix and into the white (firing) region. Since d is small, it fires as a burst ( again see part 4).

Now add the inhibitory current. It hyperpolarizes the cell, dragging the V nullcline (blue) down and moving the stable equilibrium down and to the left, away from the W nullcline (yellow). That new equilibrium is the whole trick: it lies inside the spiking basin of the baseline phase plane. While inhibition is on, the neuron sits quietly at its inhibited equilibrium, well clear of the grey resting region of the baseline plane.
Kill the inhibition and the phase plane snaps back to baseline, but the state doesn't instantaneously move back to the baseline EQ. The trajectory now finds itself on the white side of the separatrix, and off it goes. Rebound burst. Note that the inhibited equilibrium is lower in W, not just in V; the slow variable relaxes downward during the hyperpolarization. That's convenient shorthand for the biophysics below, and it also means inhibition has to actually last a while (roughly longer than 1/a) for the state to get there, a brief or weak IPSP won't do it. I won't build that out here, but it's worth keeping in the back of your mind if you're ever tuning one of these against real data.

Real Life Post-Inhibitory Rebound Currents

In real neurons the two workhorses are low-threshold T-type calcium currents and the hyperpolarization-activated cation current, I_h (HCN channels). Both are slow, and both are recruited by hyperpolarization: T-type channels de-inactivate, HCN channels activate. Sit at a negative voltage long enough and enough of them are available that releasing the inhibition lets the membrane depolarize on its own, producing a spike or a burst. Then, as the cell depolarizes during the burst, T-type inactivates and I_h shuts off, and the burst terminates on its own. Worth flagging: the Izhikevich model has one slow variable doing the job that two separate currents do biophysically, so don't lean on the mapping too hard. It's a cartoon of the mechanism, not a model of it.

That's one route to a rebound burst among many. It's just the one the Izhikevich picture maps onto most cleanly.

I've attached the code I used to make these phase planes. It is worth playing with d yourself. Push it up and you'll see the rebound collapse from a burst down to a single spike.

Author: Alexander White