Dynamical Systems in Neuroscience:

106 Two-Dimensional SystemsFor example, is the equilibrium in Fig. 4.4 stable? What about the equilibrium inFig. 4.10? The vector fields in the neighborhoods of the two equilibria exhibit subtledifferences that are difficult to spot without the help of analytical tools, which wediscuss next.4.2.2 Local linear analysisBelow we remind the reader some basic concepts of linear algebra, assuming that thereader has some familiarity with matrices, eigenvectors and eigenvalues. Consider atwo-dimensional dynamical systemẋ = f(x, y) (4.5)ẏ = g(x, y) (4.6)having an equilibrium point (x 0 , y 0 ). The nonlinear functions f and g can be linearizednear the equilibrium; i.e., written in the formf(x, y) = a(x − x 0 ) + b(y − y 0 ) + higher-order terms,g(x, y) = c(x − x 0 ) + d(y − y 0 ) + higher-order terms,where higher-order terms include (x − x 0 ) 2 , (x − x 0 )(y − y 0 ), (x − x 0 ) 3 , etc., anda = ∂f∂x (x 0, y 0 ),c = ∂g∂x (x 0, y 0 ),b = ∂f∂y (x 0, y 0 ),d = ∂g∂y (x 0, y 0 ),are the partial derivatives of f and g with respect of the state variables x and yevaluated at the equilibrium (x 0 , y 0 ) (first, evaluate the derivatives, then substitutex = x 0 and y = y 0 ). Many questions regarding the stability of the equilibrium can beanswered by considering the corresponding linear system˙u = au + bw , (4.7)ẇ = cu + dw , (4.8)where u = x − x 0 and w = y − y 0 are the deviations from the equilibrium, and thehigher-order terms, u 2 , uw, w 3 , etc., are neglected. We can write this system in thevector form( ) ( ) ( )˙u a b u=.ẇ c d wThe linearization matrix( ) a bL =c d