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Izhikevich EM (2004), Which model to use for cortical spiking neurons?

March 2011, model of the month by Junmei Zhu
Original models: BIOMD0000000129, BIOMD0000000130, BIOMD0000000131, BIOMD0000000132, BIOMD0000000133, BIOMD0000000134, BIOMD0000000135, BIOMD0000000136, BIOMD0000000141 and BIOMD0000000142

Neurons communicate mostly through action potentials, the rapid rise and fall of membrane potential. Although single spikes have a stereotypical shape, the temporal firing patterns vary tremendously.

20 most prominent patterns of biological spiking neurons are shown in figure 1 (taken from [1]). Having a model being able to exhibit all or most of these patterns is important when the goal is to understand whether and how cortical information processing uses the fine temporal structure of spike trains.

Models of cortical spiking neurons, conveniently expressed in ordinary differential equations, are abundant. Figure 2 (taken from [1]) compares a number of models in terms of how well they exhibit various spiking patterns and their computational complexity for simulation.

The Hodgkin-Huxley model [2, BIOMD0000000020], which can produce all the patterns, represents one of the greatest achievements of biological modeling. For this work, Hodgkin and Huxley received the 1963 Nobel Prize in Physiology or Medicine. The dynamics are later simplified in the Fitzhugh-Nagumo model for analysis. Notable further development includes the Morris-Lecar model and the Hindmarsh-Rose model. In these models, the action potential rises from the continuous dynamics, therefore in simulation the time step has to be small.

Another group of models use explicit thresholding and resetting to generate action potential, which are algorithmic but efficient in simulation. Examples include the most commonly used integrate-and-fire (I&F) model, and its modifications such as I&F with adaptation, I&F or burst, and quadratic I&F. This type of models are very limited in the patterns they can exhibit, with one exception shown in this model of the month.

The model by Izhikevich 2003 [3, BIOMD0000000127], is the simplest possible model that can exhibit all the firing properties in figure 1.

The dynamic read

v' = 0.04v2 + 5v +140 - u + I → (eqn 1)

u' = a(bv - u) → (eqn 2)

with the auxiliary after-spike resetting:

if v >= 30 mV, then v ← c, u ← u +d

where v is the membrane potential of the neuron in mV and u is a membrane recovery variable. Time has ms scale.

Figure 1

Figure 1: Summary of the neuro-computational properties of biological spiking neurons. Figure taken from [1] and

Figure 2

Figure 2: Comparison of the neuro-computational properties of spiking and bursting models. Figure taken from [1].

Izhikevich 2004 [1, BIOMD0000000129 - BIOMD0000000136 and BIOMD0000000141 - BIOMD0000000142] review important neuro-computational features of real neurons and their contribution to temporal coding and spike-timing information processing. All the responses in figure 1 are obtained using the above described model proposed by Izhikevich 2003 [3]. With the choice of parameters a, b, c and d, the model can exhibit firing patterns of all known types of cortical neurons. In particular, these are the corresponding BioModels Database identifiers of the model corresponding to each of the plot in figure 1.

BioModels Database model identifiers (plot in figure 1):

This model has been used successfully in real-time simulation with 1ms resolution of large networks (say of 100,000 spiking neurons).

In summary, this Izhikevich 2003 is the model of choice, if one needs all or most of the spiking patterns and to simulate large-scale (thousands) spiking neuronal networks in real-time. If the goal is to study neural behaviour that depends on physiological parameters, on the other hand, then use the Hodgkin-Huxley model, keeping in mind that only a small number (tens) of coupled neurons can be simulated in real-time.

Bibliographic References

  1. Izhikevich EM. Which model to use for cortical spiking neurons? IEEE Trans Neural Netw , 15(5): 1063-70, 2004. [CiteXplore]
  2. Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. , 117(4): 500-44, 1952. [CiteXplore]
  3. Izhikevich EM. Simple model of spiking neurons. IEEE Trans Neural Netw , 14(6): 1569-72, 2003. [CiteXplore]