Dynamical Systems in Neuroscience:
Dynamical Systems in Neuroscience: Dynamical Systems in Neuroscience:
94 Two-Dimensional SystemsI Na,pneuronI KI0high-thresholdI KI(V)=I L +I Na,p +I KILa bcurrent (pA)100-1000low-thresholdI(V)=I L +I Na,p +I KI K-100 E -50 0 50KE Namembrane potential, V (mV)-100E -50 0 50KE Namembrane potential, V (mV)Figure 4.1: The I Na,p +I K -model (4.1, 4.2). Parameters in (a): C = 1, I = 0, E L = −80mV, g L = 8, g Na = 20, g K = 10, m ∞ (V ) has V 1/2 = −20 and k = 15, n ∞ (V ) hasV 1/2 = −25 and k = 5, and τ(V ) = 1, E Na = 60 mV and E K = −90 mV. Parametersin (b) as in (a) except E L = −78 mV and n ∞ (V ) has V 1/2 = −45; see Sect. 2.3.5.Figure 4.2: Harold Lecar (back), RichardFitzHugh (front), and Cathy Morris atNIH Biophysics Lab, summer of 1983.
Two-Dimensional Systems 9510886644y20y20-2-2-4-4-6-6-8-8-1010-10 -8 -6 -4 -2 0 2 4 6 8 10xa-10-10 -8 -6 -4 -2 0 2 4 6 810xb886644y20-2-4-6-8-10-10 -8 -6 -4 -2 0 2 4 6 8 10xcy20-2-4-6-8-10-10 -8 -6 -4 -2 0 2 4 6 8 10xdFigure 4.3: Examples of vector fields.indicates the direction of change of the state variable. For example, negative f(x 0 , y 0 )and positive g(x 0 , y 0 ) imply that x(t) decreases and y(t) increases at this particularpoint. Since each point on the phase plane (x, y) has its own vector (f, g), the systemabove is said to define a vector field on the plane, also known as direction field or velocityfield, see Fig. 4.3. Thus, the vector field defines the direction of motion; depending onwhere you are, it tells you where you are going.Let us consider a few examples. The two-dimensional systemẋ = 1 ,ẏ = 0defines a constant horizontal vector field in Fig. 4.3a since each point has a horizontalvector (1, 0) attached to it. (Of course, we depict only a small sample of vectors.)Similarly, the systemẋ = 0 ,ẏ = 1
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Two-Dimensional <strong>Systems</strong> 9510886644y20y20-2-2-4-4-6-6-8-8-1010-10 -8 -6 -4 -2 0 2 4 6 8 10xa-10-10 -8 -6 -4 -2 0 2 4 6 810xb886644y20-2-4-6-8-10-10 -8 -6 -4 -2 0 2 4 6 8 10xcy20-2-4-6-8-10-10 -8 -6 -4 -2 0 2 4 6 8 10xdFigure 4.3: Examples of vector fields.<strong>in</strong>dicates the direction of change of the state variable. For example, negative f(x 0 , y 0 )and positive g(x 0 , y 0 ) imply that x(t) decreases and y(t) <strong>in</strong>creases at this particularpo<strong>in</strong>t. S<strong>in</strong>ce each po<strong>in</strong>t on the phase plane (x, y) has its own vector (f, g), the systemabove is said to def<strong>in</strong>e a vector field on the plane, also known as direction field or velocityfield, see Fig. 4.3. Thus, the vector field def<strong>in</strong>es the direction of motion; depend<strong>in</strong>g onwhere you are, it tells you where you are go<strong>in</strong>g.Let us consider a few examples. The two-dimensional systemẋ = 1 ,ẏ = 0def<strong>in</strong>es a constant horizontal vector field <strong>in</strong> Fig. 4.3a s<strong>in</strong>ce each po<strong>in</strong>t has a horizontalvector (1, 0) attached to it. (Of course, we depict only a small sample of vectors.)Similarly, the systemẋ = 0 ,ẏ = 1
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