Difference between revisions of "FitzHugh-Nagumo System"

Revision as of 17:37, 20 November 2012

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

$\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,$

$w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,$

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

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

$v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,$

where $\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05,\;i_0(t)=5\cdot 10^4t^3 \exp(-15t).$ 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

Contact information:

Tobias Breiten

@@ 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==