What is the cause of rebound spikes?

I am solving this question

Inject a hyperpolarizing step current I_amp = -5 uA for 20ms into the HH neuron. Simulate the neuron for 50 ms and plot the voltage trace and the gating variables. What is happening here? To which gating variable do you attribute this rebound spike?

using brian2 and neurodynex3.

step_cur = input_factory.get_step_current(5, 25, b2.ms, -5* b2.uA)
state_monitor = HH.simulate_HH_neuron(step_cur, 50 * b2.ms)
HH.plot_data(state_monitor,title="HH Neuron")

Screenshot from 2021-10-07 21-14-12

I am unable to find the explanation of rebound spike in simple terms anywhere.

I really like the online textbook Neuronal Dynamics,
it has a relevant section for your question:
2.3.5 Calcium spikes and postinhibitory rebound

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Hi @willtryagain,

That is an excellent question, indeed!
Although the general idea of how rebound spike is generated is well described in the chapter @adam mentioned, you have quite a different story here. This PIR spike appears because the HH model has Type-2 excitability (so-called resonator), and it is based on the tight interplay between sodium m and potassium n variables. This type of excitability and (potentially) fast PIR spikes are often seen in cortical PV+ basket cells.

If you look closer, you can find that at rest, sodium channels are a bit open, which is well compensated by potassium channels, which are a bit open too. When you hyperpolarized the HH model, both channels close. However, then you suddenly release hyperpolarization, m-variable recovers faster than n and runs into the spike. The HH model, as well as any type 2 excitability, generates PIR spike much easily if you slightly depolarize it. Indeed, real PV+ neurons do the same! Here a figure from my 2015 paper:

We argue that these PIR spikes allow type 2 to be network oscillators, even when excitation is insufficient to keep neurons firing. Moreover, the network oscillates in gamma frequencies because the dynamics of these fast PIR spikes are different from those of calcium PIR spikes.

I recommend reading at least the first and the last papers in the list below to understand how these PIR spikes can provide exceptionally robust and reliable mechanisms for gamma rhythm oscillations.

  1. Tikidji-Hamburyan R.A. , Martínez J.J., White J.A., Canavier C.C. (2015) Resonant Interneurons Can Increase Robustness of Gamma Oscillations J Neuroscience 35(47): 15682-15695
  2. Canavier C.C., Tikidji-Hamburyan R.A. (2017) Globally Attracting Synchrony in a Network of Oscillators with All-to-All Inhibitory Pulse Coupling Phys. Rev. E 95 , 032215
  3. Tikidji-Hamburyan R.A. ,Leonik C.A., Canavier C.C. (2019) Phase Response Theory Explains Cluster Formation in Sparsely but Strongly Connected Inhibitory Neural Networks and Effects of Jitter due to Sparse Connectivity J Neurophysiol 121 : 1125–1142
  4. Tikidji-Hamburyan R.A. , Canavier C.C. (2020) Shunting Inhibition Improves Synchronization in Heterogeneous Inhibitory Interneuronal Networks with Type 1 Excitability Whereas Hyperpolarizing Inhibition is Better for Type 2 Excitability eNeuro 7(3) ENEURO.0464-19.2020 1–22

Hope it helps. You are welcome to ask any questions.
An example of a Brian2 model with these PIR gamma oscillations can be find here

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