Dynamical Systems in Neuroscience:

Introduction 7Figure 1.8: Neurons are dynamical systems.currents, say I Na,p and I K , and exhibiting excitable behavior; that is, both neurons arequiescent but can fire a spike in response to a stimulation. Suppose the second neuronhas stronger I Na,p , which is balanced by stronger I K . If we measure Na + conductanceusing the first neuron and K + conductance using the second neuron, the resultingI Na,p + I K -model would have an excess of K + current and probably not be able to firespikes at all. Conversely, if we measure Na + and K + conductances using the secondand then the first neuron, respectively, the model would have too much Na + currentand probably exhibit sustained pacemaking activity. In any case, the model fails toreproduce the excitable behavior of the neurons whose parameters we measured.Some of the parameters cannot be measured at all, so many arbitrary choices aremade via a process called “fine-tuning”. Navigating in the dark, possibly with the helpof some biological intuition, the researcher modifies parameters, compares simulationswith experiment, and repeats this trial-and-error procedure until he or she is satisfiedwith the results. Since seemingly similar values of parameters can result in drasticallydifferent behaviors, and quite different parameters can result in seemingly similar behaviors,how do we know that the resulting model is correct? How do we know that itsbehavior is equivalent to that of the neuron we want to study? And what is equivalentin this case? Now, the reader is primed to consider dynamical systems.1.2 Dynamical SystemsIn the next chapter we introduce the Hodgkin-Huxley formalism to describe neuronaldynamics in terms of activation and inactivation of voltage-gated conductances. Animportant consequence of the Hodgkin-Huxley studies is that neurons are dynamicalsystems, so they should be studied as such. Below we mention some of the important