Dynamical Systems in Neuroscience:
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Dynamical Systems in Neuroscience:

Dynamical Systems in Neuroscience: Dynamical Systems in Neuroscience:

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126 Two-Dimensional SystemsReview of Important Concepts• A two-dimensional system of differential equationsẋ = f(x, y)ẏ = g(x, y) ,describes joint evolution of state variables x and y, which often are the membranevoltage and a “recovery” variable.• Solutions of the system are trajectories on the phase plane R 2 that are tangentto the vector field (f, g).• The sets given by the equations f(x, y) = 0 and g(x, y) = 0 are the x- andy-nullclines, respectively, where trajectories change their x and y directions.• Intersections of the nullclines are equilibria of the system.• Periodic dynamics correspond to closed loop trajectories.• Some special trajectories, e.g., separatrices, define thresholds and separateattraction domains.• An equilibrium or a periodic trajectory is stable if all nearby trajectories areattracted to it.• To determine the stability of an equilibrium, one needs to consider the Jacobianmatrix of partial derivatives( )fx fL =y.g x g y• The equilibrium is stable when both eigenvalues of L are negative or havenegative real parts.• The equilibrium is a saddle, a node, or a focus, when L has real eigenvaluesof opposite signs, of the same signs, or complex-conjugate eigenvalues,respectively.• When the equilibrium undergoes a saddle-node bifurcation, one of the eigenvaluesbecomes zero.• When the equilibrium undergoes an Andronov-Hopf bifurcation (birth or deathof a small periodic trajectory) the complex-conjugate eigenvalues becomepurely imaginary.• The saddle-node and Andronov-Hopf bifurcations are ubiquitous in neuralmodels, and they result in different neuro-computational properties.

Two-Dimensional Systems 127Bibliographical NotesAmong many textbooks on the mathematical theory of dynamical systems we recommendthe following three:• Nonlinear Dynamics and Chaos by Strogatz (1994) is suitable as an introductorybook for undergraduate math or physics majors or graduate students in lifesciences. It contains many exercises and worked out examples.• Differential Equations and Dynamical Systems by Perko (1996, Third editionin 2000) is suitable for math and physics graduate students, but may be tootechnical for life scientists. Nevertheless, it should be a standard textbook forcomputational neuroscientists.• Elements of Applied Bifurcation Theory by Kuznetsov (1995, Third edition in2004) is suitable for advanced graduate students in mathematics or physics andfor computational neuroscientists who want to pursue bifurcation analysis of neuralmodels.The second edition of The Geometry of Biological Time by Winfree (2001) is a goodintroductory book into oscillations, limit cycles, and synchronization in biology. Itrequires little background in mathematics and can be suitable even for undergraduatelife science majors. Mathematical Biology by Murray (1993, Third edition in 2003) is anexcellent example how dynamical system theory can solve many problems in populationbiology and shed light on pattern formation in biological systems. Most of this book issuitable for advance undergraduate or graduate students in mathematics and physics.Mathematical Physiology by Keener and Sneyd (1998) is similar to Murray’s book, butis more focused on neural systems. Spikes, Decisions, and Actions by Wilson (1999)is a short introduction to dynamical systems with many neuroscience examples.Exercises1. Use pencil (as in Fig. 4.39) to sketch the nullclines of the vector fields depictedin figures 4.40 through 4.44.2. Assume that the continuous curve is the x-nullcline and the dashed curve is they-nullcline in Fig. 4.38, and that ẋ or ẏ changes sign when (x, y) passes throughthe corresponding nullcline. The arrow indicates the direction of the vector fieldin one region. Determine the approximate directions of the vector field in theother regions of the phase plane.3. Use pencil (as in Fig. 4.39) to sketch phase portraits of the vector fields depictedin figures 4.40 through 4.44. Clearly mark all equilibria, their stability, andattraction domains. Show directions of all homoclinic, heteroclinic and periodic

Two-Dimensional <strong>Systems</strong> 127Bibliographical NotesAmong many textbooks on the mathematical theory of dynamical systems we recommendthe follow<strong>in</strong>g three:• Nonl<strong>in</strong>ear Dynamics and Chaos by Strogatz (1994) is suitable as an <strong>in</strong>troductorybook for undergraduate math or physics majors or graduate students <strong>in</strong> lifesciences. It conta<strong>in</strong>s many exercises and worked out examples.• Differential Equations and <strong>Dynamical</strong> <strong>Systems</strong> by Perko (1996, Third edition<strong>in</strong> 2000) is suitable for math and physics graduate students, but may be tootechnical for life scientists. Nevertheless, it should be a standard textbook forcomputational neuroscientists.• Elements of Applied Bifurcation Theory by Kuznetsov (1995, Third edition <strong>in</strong>2004) is suitable for advanced graduate students <strong>in</strong> mathematics or physics andfor computational neuroscientists who want to pursue bifurcation analysis of neuralmodels.The second edition of The Geometry of Biological Time by W<strong>in</strong>free (2001) is a good<strong>in</strong>troductory book <strong>in</strong>to oscillations, limit cycles, and synchronization <strong>in</strong> biology. Itrequires little background <strong>in</strong> mathematics and can be suitable even for undergraduatelife science majors. Mathematical Biology by Murray (1993, Third edition <strong>in</strong> 2003) is anexcellent example how dynamical system theory can solve many problems <strong>in</strong> populationbiology and shed light on pattern formation <strong>in</strong> biological systems. Most of this book issuitable for advance undergraduate or graduate students <strong>in</strong> mathematics and physics.Mathematical Physiology by Keener and Sneyd (1998) is similar to Murray’s book, butis more focused on neural systems. Spikes, Decisions, and Actions by Wilson (1999)is a short <strong>in</strong>troduction to dynamical systems with many neuroscience examples.Exercises1. Use pencil (as <strong>in</strong> Fig. 4.39) to sketch the nullcl<strong>in</strong>es of the vector fields depicted<strong>in</strong> figures 4.40 through 4.44.2. Assume that the cont<strong>in</strong>uous curve is the x-nullcl<strong>in</strong>e and the dashed curve is they-nullcl<strong>in</strong>e <strong>in</strong> Fig. 4.38, and that ẋ or ẏ changes sign when (x, y) passes throughthe correspond<strong>in</strong>g nullcl<strong>in</strong>e. The arrow <strong>in</strong>dicates the direction of the vector field<strong>in</strong> one region. Determ<strong>in</strong>e the approximate directions of the vector field <strong>in</strong> theother regions of the phase plane.3. Use pencil (as <strong>in</strong> Fig. 4.39) to sketch phase portraits of the vector fields depicted<strong>in</strong> figures 4.40 through 4.44. Clearly mark all equilibria, their stability, andattraction doma<strong>in</strong>s. Show directions of all homocl<strong>in</strong>ic, heterocl<strong>in</strong>ic and periodic

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