Izhikevich EM (2003), Simple model of spiking neurons.
May 2010, model of the month by Michele Mattioni
Original model: BIOMD0000000127
Modeling is always a trade off between abstraction and reality; in computational neurobiology there is another trade off to make: computational power and models' details.
In 1952, Hodgkin and Huxley [1] proposed a mathematical formalism [BIOMD0000000020], to model ionic channels, where the biochemical gates are explicitly expressed. This system offers an accurate way to investigate the properties of the channels in a small amount of neurons. However, this cannot be used for a larger network of neurons due to the demanding computational power.
The integrate and fire neuron [2,3] formalism has been used to model very large networks of neurons, as they are not computationally demanding. However, this formalism is not capable of simulating any kind of complex firing patterns of a single neuron.
Izhikevich 2003 [4, BIOMD0000000127], in this paper, proposed a different kind of formalism which is able to replicate different rich firing patterns, using two simple equations with only one supralinear term.
The equations are:
v' = 0.04v^{2} + 5v +140  u + I → (eqn 1)
and
u' = a(bv  u) → (eqn 2)
with an auxiliary to reset the voltage after the spike to the resting potential:
if v = 30 mV, then v ← c, u ← u +d
The four parameters: a, b, c and d have completely different roles. The parameter "a" represents the recovery of the membrane after the spike, the parameter "b" takes into account the sensibility of the neuron to the fluctuation of the voltage, the parameter "c" is used to set the maximum amplitude of the spike after which the neuron's voltage is reset and the parameter "d" determines the afterspike overshoot reset.

Figure 1: Known types of neurons correspond to different values of the parameters a, b, c, d in the model described by the equations (eqn 1) and (eqn 2). RS, IB, and CH are cortical excitatory neurons. FS and LTS are cortical inhibitory interneurons. Each inset shows a voltage response of the model neuron to a step of decurrent I = 10 (bottom). Time resolution is 0.1ms. Figure taken from [4].

Figure 2: Simulation of a network of 1000 randomly coupled spiking neurons. Top: spike raster shows episodes of alpha and gamma band rhythms (vertical lines). Bottom: typical spiking activity of an excitatory neuron. All spikes were equalized at +30mV by resetting v1 first to +30mV and then to c. Figure taken from [4].

The author shows in Figure 1, the possibility to replicate the firing patterns of some of the known neurons in the cortex just by changing the parameters in a proper way. The neurons varies from the excitatory neurons  like the Regular Spiking (RS), to the inhibitory ones  like the Fast Spiking Interneurons (FS).
The model was used to run a simulation of 10000 spiking cortical neurons with 1000000 synaptic connections using only a 1Ghz desktop PC and C++. The result of this simulation is shown in Figure 2. The model is able to replicate known types of cortical states like the alpha and gamma wave. Other known states of the cortex, like sleep oscillation or spindle wave can be produced by changing the synaptic strength and the thalamic drive of the model.
In 2008, Izhikevich et al. [5] used this formalism in a largescale model of mammalian thalamocortical systems, where compartmental neurons where used. Although this formalism is not too demanding from a computational point of view, the model ran for 50 days on 27 CPUs cluster to simulate 1 second.
In conclusion, the formalism presented in this paper by Izhekivich is the first model being able to simulate large scale network neurons with rich firing patterns with a less computational cost.

Bibliographic References

Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. , 117(4): 50044, 1952. [CiteXplore]

Lapicque L. Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarization. J. Physiol. Pathol. Gen. , 9: 620635, 1907. [Wikipedia]

Abbott LF. Lapique's introduction of the integrateandfire model neuron (1907). Brain Res Bull. , 50(56): 3034, 1999. [CiteXplore]

Izhikevich EM. Simple model of spiking neurons. IEEE Trans Neural Netw , 14(6): 156972, 2003. [CiteXplore]

Izhikevich EM, Edelman GM. Largescale model of mammalian thalamocortical systems. Proc Natl Acad Sci U S A., 105(9): 35938, 2008. [CiteXplore]
