Dynamical Systems in Neuroscience:

Simple Models 289clever numerical methods are needed to catch the exact moment of threshold crossing(Hansel et al. 1998). In contrast, the slope is nearly infinite in the simple model, sono special methods are needed to identify the peak of the spike.In Fig. 8.8 we used the model to reproduce the 20 of the most fundamental neurocomputationalproperties of biological neurons. Let us check that the model is thesimplest possible system that can exhibit the kind of behavior in the figure. Indeed,it has only one non-linear term, i.e., v 2 . Removing the term makes the model linearand equivalent to the resonate-and-fire neuron (though it becomes analytically solvable).Removing the recovery variable u makes the model equivalent to the quadraticintegrate-and-fire neuron with all its limitations, such as inability to burst or to bea resonator. In summary, we found the simplest possible model capable of spiking,bursting, being an integrator or a resonator, and it should be the model of choice insimulations of large-scale networks of spiking neurons.8.1.5 Canonical modelsIt is quite rare, if ever possible, to know precisely the parameters describing dynamicsof a neuron (many erroneously think that the Hodgkin-Huxley model of squid axon isan exception). Indeed, even if all ionic channels expressed by the neuron are known,the parameters describing their kinetics are usually obtained via averaging over manyneurons; there are measurement errors; the parameters change slowly, etc. Thus, we areforced to consider families of neuronal models with free parameters, e.g. the family ofI Na +I K -models. It is more productive from computational neuroscience point of viewto consider families of neuronal models having a common property, e.g., the family ofall integrators, the family of all resonators, or the family of “fold/homoclinic” burstersconsidered in the next chapter. How can we study the behavior of the entire family ofneuronal models if we have no information about most of its members?The canonical model approach addresses this issue. Briefly, a model is canonicalfor a family if there is a piece-wise continuous change of variables that transforms anymodel from the family into this one, as we illustrate in Fig. 8.10. The change of variablesdoes not have to be invertible, so the canonical model is usually lower-dimensional,simple, and tractable. Yet, it retains many important features of the family. Forexample, if the canonical model has multiple attractors, then each member of thefamily has multiple attractors. If the canonical model has a periodic solution, theneach member of the family has a periodic (quasi-periodic or chaotic) solution. If thecanonical model can burst, then each member of the family can burst. The advantageof this approach is that we can study universal neuro-computational properties thatare shared by all members of the family since all such members can be put into thecanonical form by a change of variables. Moreover, we need not actually present sucha change of variables explicitly, so derivation of canonical models is possible even whenthe family is so broad that most of its members are given implicitly, e.g., the family of“all resonators”.The process of deriving canonical models is more an art than a science, since a