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Dynamical Systems in Neuroscience:

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86 One-Dimensional <strong>Systems</strong>spike right after the bifurcation. This type of behavior is typical <strong>in</strong> sp<strong>in</strong>y projectionneurons of neostriatum and basal ganglia, as we show <strong>in</strong> Chap. 8.Review of Important Concepts• The one-dimensional dynamical system ˙V = F (V ) describes howthe rate of change of V depends on V . Positive F (V ) means V<strong>in</strong>creases, negative F (V ) means V decreases.• In the context of neuronal dynamics, V is often the membrane potential,and F (V ) is the steady-state I-V curve taken with the m<strong>in</strong>ussign.• A zero of F (V ) corresponds to an equilibrium of the system. (Indeed,if F (V ) = 0, then the state of the system, V , neither <strong>in</strong>creases nordecreases.)• An equilibrium is stable when F (V ) changes the sign from “+” to“−”. A sufficient condition for stability is that the eigenvalue λ =F ′ (V ) at the equilibrium be negative.• A phase portrait is a geometrical representation of the system’s dynamics.It depicts all equilibria, their stability, representative trajectories,and attraction doma<strong>in</strong>s.• A bifurcation is a qualitative change of the system’s phase portrait.• The saddle-node (fold) is a typical bifurcation <strong>in</strong> one-dimensionalsystems: As a parameter changes, a stable and an unstable equilibriumapproach, coalesce, and then annihilate each other.Bibliographical NotesThere is no standard textbook on dynamical systems theory. The classical book Nonl<strong>in</strong>earOscillations, <strong>Dynamical</strong> <strong>Systems</strong>, and Bifurcations of Vector Fields by Guckenheimerand Holmes (1983) plays the same role <strong>in</strong> the dynamical systems communityas the book Ion Channels of Excitable Membranes by Hille (2001) <strong>in</strong> the neurosciencecommunity. A common feature of these books is that they are not suitable as a firstread<strong>in</strong>g on the subject.Most textbooks on differential equations, such as Differential Equations and <strong>Dynamical</strong><strong>Systems</strong> by Perko (1996), develop the theory start<strong>in</strong>g with a comprehensiveanalysis of l<strong>in</strong>ear systems, then apply<strong>in</strong>g it to local analysis of non-l<strong>in</strong>ear systems, andthen discuss<strong>in</strong>g global behavior. By the time the reader gets to bifurcations, he hasto go through a lot of daunt<strong>in</strong>g math, which is fun only for mathematicians. Here we

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