where Cij is the matrix of coupling coefficients. This is called gap junctional, electrical or diffusive coupling. For most neural models only the (1,1) element of Cij is non zero.
There are several sources of delay in a neural system. Consider first the delay due to propagation of action potentials along the axon. In the model above, when an action potential is generated in the cell body of neuron j, it is immediately felt by all other neurons to which it is connected. However, in reality, the action potential must travel along the axon of neuron j to the synapse or gap junction. Conduction velocities can range from the order of 1 m/sec along unmyelinated axons to more than 100 m/sec along myelinated axons (Desmedt and Cheron, 1980; Shepherd, 1994). This can lead to significant time delays in certain brain structures. There are several ways to incorporate this into the model, such as including spatial dependence of the variables or multiple compartments representing different parts of the neuron (Koch, 1990). However, if we are primarily interested in the effect of the action potential when it reaches the end of the axon (will it cause an action potential in another neuron?), then a simpler approach is to include a time delay in the coupling term. In this case the general coupling term becomes fij (xi^ xj (t — Tij )) (4)
where Tij > 0 represents the time taken for the action potential to propagate along the axon connecting neuron j (the pre-synaptic neuron) to neuron i (the post-synaptic neuron).
The above assumes that the axon of neuron j connects on or close to the cell body of neuron i. Some cells may have synapses or gap junctions on dendrites far from the cell body. In this case, there can also be a delay associated with propagation of the action potential along the dendrite. This will introduce an additional time delay, viz., fij (x (t — Tdj ), xj (t — Tdj — Tij )) (5)
where Tij and Tdj represent the time delays due to the propagation of the action potential along the axon and dendrite, respectively.
Another delay can occur in the transmission of the signal across the synapse. That is, once the action potential from neuron j reaches the synapse, there is some time before an action potential is initiated in neuron i. A common way to model this is to augment the model equations above by equations modelling the chemical kinetics of the synapse (Keener and Sneyd, 1998; Koch, 1999). Alternatively, this can be incorporated into (4) or (5) just by increasing the delay Tij. I will focus on the latter approach, but in Sect. 3 will review some literature that indicates the qualitative effect on the dynamics can be quite similar using both approaches. Clearly, the latter approach will yield a simpler model if one also wants to include the effect of axonal delay.
Equations (4) and (5) assume that the time delays are fixed. In reality, the delay will likely vary slightly each time an action potential is propagated from neuron j to neuron i. This may be incorporated into the model putting time dependence into the delay: Tij (t). Many of the methods outlined in Sect. 2 may be extended to this case, by assuming the delay satisfies some constraints 0 < Tij (t) < fij. Alternatively, one might consider adding some noise to the delay, which would lead to a stochastic delay differential equation model. Unfortunately, there is very little theory available for such equations.
An alternative approach is to incorporate a distribution of delays, representing the situation where the delay occurs in some range of values with some associated probability distribution. In this case, coupling term (4) becomes
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