The Izhikevich Neuron Model
Paper Summary: “Simple Model of Spiking Neurons”
Paper by: Eugene M. Izhikevich
Side Note — Paper Implementation:
I implemented the Izhikevich Neuron Model! You can check out my implementation of the network (with all the synaptic connections) here! I also implemented the neuron model itself with all the different neural dynamics! You can check it out here!
Abstract:
This paper presents a paper that takes the biological plausibility of Hodgkin and Huxley neurons while being as computationally efficient as integrate-and-fire neurons. It can reproduce the spiking and bursting of cortical neurons + we can simulate thousands of them on a regular computer.
Introduction:
If we want to study how the brain works, we have to simulate realistic neurons on large scales. Thus we must compromise between accuracy and computational efficiency.
Hodgkin and Huxley's neurons are on one extreme. They’re very accurate, but we can only simulate a handful of these neurons since it’s so computationally expensive. On the other hand, we have integrate-and-fire neurons. These are computationally effective, but it’s unrealistic simple — thus not able to reproduce rich spiking and bursting that’s in real neurons.
This paper takes the best parts of both models and combines them. It’s as biologically plausible as Hodkin and Huxley neurons, while also being as computationally efficient as integrate-and-fire neurons. There are only 4 parameters to tune, and we can reproduce the complex behaviour of cortical neurons.
The Model:
We can represent this model through a 2-D system of differential equations:
- v and u = dimensionless variables
- a, b, c, d = dimensionless variables
- ‘ = d/dt → Derivative with respect to time
- v = membrane potential
- u = membrane recovery variable
- I = synpatic currents / injected dc-currents
In biological neurons, the Na^+ ions enter into the neuron to increase the voltage of the neuron. At a certain point, the floodgates open and Na^+ ions come rushing into the neuron.
This causes the voltage inside the neuron to shoot right up (spike). But it’s this point where the K^+ gates open up, and a ton of K^+ ions shoot out of the neuron → Causing the voltage to drop right down
The voltage in this case is the membrane potential (v). And the K^+ is the membrane recovery variable (u). The reset follows this formula:
Where did the 0.04v² + 5v + 140 come from? It was obtained by fitting the spike initiation dynamics of real neurons, such that v is in mV and t is in ms. The resting potential is between -70 and -60 mV. There is no fixed threshold (like real neurons) which can be between -55 and -40 mV
- a = time scale of recovery variable u. Magnitude is proportional to recovery speed (small = slower recovery time). A typical value of 0.02
- b = sensitivity of u to the fluctuations in v. The bigger the value, the more their coupled, meaning the more there are subthreshold oscillations and low threshold spiking. A typical value of 0.2.
- b < a = saddle-node bifurcation in the resting state
- b > a = Andronov-Hopf bifurcation in the resting state
- c = after-spike reset value of v. A typical value of -65mV
- d = after-spike reset value of u. A typical value of 2
We could extend this (by making u, a, and b as vectors). But it’s unnecessary. The variables that we have are enough to reproduce neocortical and thalamic neuron activities!
Different Types of Dynamics:
There are lots of types of patterns spiking and bursting inside of cells! Below are the descriptions of the different types:
Excitatory Cortical Cells:
Regular Spiking (RS) neurons — These are your normal neurons. Prolonged stimuli cause the neurons to fire a few times in short periods, and the periods increase. It has spike frequency adaptation. If we increase the injected dc-current, we also increase the frequency, but it’s never too fast because we have long after hyperpolarization. c = -65 mV (deep), d = 8 (large)
Intrinsically Bursting (IB) neurons — First, we fire a ton of bursts, but after a while, we end up with normal spiking. This is because the u builds up over time. c = -55 mV (high), d = 4 (large)
Chattering (CH) neurons — We fire spaced apart bursts. The frequency inside of bursts can be as high as 40 Hz. c = -50 mV (high), d = 2 (moderate)
Inhibitory Cortical Cells:
Fast Spiking (FS) neurons — Can fire spikes super quickly at high frequencies with 0 slowing down. a = 0.1 (fast)
Low-Threshold Spiking neurons — As the name suggests, it’s spiking at low thresholds. The intervals increase with respect to time. b = 0.25 (low)
Thalamo-Cortical Neurons:
Thalamo-Corical (TC) neurons — This neuron has 2 modes — When it’s at rest and then depolarized it ends up like the left diagram.
But if we apply a negative current step, then the membrane potential is hyperpolarized (voltage decreases to -90 mV) and the neurons will fire rebound bursts. This ends up like the right diagram.
Resonator (RZ) neurons — in here there are sustained oscillations around the subthreshold. There’s bistability of the resting and repetitive striking states which we can switch between. a = 0.1, b = 0.26
And the list goes on and on. The Izhikevich Neuron Model is a one-size-fits-all model because it can simulate lots of different types of neurons. The Izhikevish model is for all these neurons, but if we wanted a specific type of neuron you can modify the differential equation for the voltage to get more specific optimal neurons.
Pulse-Coupled Implementation:
We’re able to simulate a network on 10,000 spiking neurons with 1,000,000 synaptic connections at a resolution of 1ms on a 1 GHz PC. {Keep in mind that this paper was released in 2003. 1,000,000 parameters back then must have been crazy big}. Here’s the code to simulate it:
I implemented this Matlab code in Python using NumPy! You can play around with it here!
I also did it for a singular neuron to try and reproduce the results of the paper — I was able to do it for all of them except Intrinsically Bursting And Resonator. You can play around with the different dynamics here!
Back to the paper — so the implementations of the neural model were literally: excitatory and inhibitory neurons that were connected randomly — yet we can observe some interesting characteristics:
We’re able to observe characteristics like Poisson Spike trains (neurons that fire every 8 Hz). But we can also see dark lines — meaning that there are synchronized firings of the neurons!
They’re connected randomly, with no synaptic plasticity — yet they’re able to exhibit complex behaviour! We can change the strength of the synaptic connections, and this can produce other behaviours.
We’re able to observe all these known behaviours of biological neurons because the Izhikevich model can accurately emulate the biological neuron! Thus there isn’t a gap between biological and computational neurons!
Conclusion:
This paper proposes the Izhikevish model! It’s a simple model (2D) that’s able to reproduce complex behaviour observed in biological neurons. This neuron model has the biological plausibility of a 4D model, yet has the computational efficiency of a 1D neuron. The Izhikevish model is awesome.
If you want to find out more: Read the paper here!
Thanks for reading! I’m Dickson, an 18-year-old ML enthusiast who’s excited to use it to impact billions of people 🌎
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