FitzHugh-Nagumo System: Difference between revisions

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Description

The FitzHugh-Nagumo system describes a prototype of an excitable system (e.g., a neuron). If the external stimulus $i_{0} (t)$ exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables $v$ and $w$ relax back to their rest values. This behaviour is typical for spike generations (=short elevation of membrane voltage $v$ ) in a neuron after stimulation by an external input current.

Here, we present the setting from [1], where the equations for the dynamical system read

$ϵ v_{t} (x, t) = ϵ^{2} v_{x x} (x, t) + f (v (x, t)) - w (x, t) + g,$

$w_{t} (x, t) = h v (x, t) - γ w (x, t) + g,$

with $f (v) = v (v - 0.1) (1 - v)$ and initial and boundary conditions

$v (x, 0) = 0, w (x, 0) = 0, x \in [0, 1],$

$v_{x} (0, t) = - i_{0} (t), v_{x} (1, t) = 0, t \geq 0,$

where $ϵ = 0.015, h = 0.5, γ = 2, g = 0.05, i_{0} (t) = 5 \cdot 1 0^{4} t^{3} \exp (- 15 t) .$ In [1], the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with $k = 512$ nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension $n = 1024 .$

File:FHN.jpg

References

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@@ Line 12: / Line 12: @@
 <math>
-\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g, \\
+\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
 </math>
@@ Line 18: / Line 18: @@
   w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
 </math>
+with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions
+<math>
+  v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
+</math>
+<math>
+  v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
+</math>
+where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05,\;i_0(t)=5\cdot
+^4t^3 \exp(-15t).</math> In [1], the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>n=1024. </math>
+[[File:FHN.jpg]]
 ==References==