FitzHugh-Nagumo System

Description

The FitzHugh-Nagumo system describes a prototype of an excitable system, e.g., a neuron. If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

Figure 1: FitzHugh-Nagumo System

Model Equations

Here, we present the setting from , 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 $\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot 10^4t^3 \exp(-15t)$ is the external stimulus, and the variables $v$ and $w$ are the voltage and the recovery of the voltage, respectively. xx--CrossReference--dft--fig:fhn--xx shows the typical limit cycle behaviour described above.

Reformulation as a quadratic-bilinear system

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

$E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),$

with $E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2}$ and $B \in \mathbb R^{n\times 2}.$ The idea relies on artificially introducing a new state variable defined as $z(t)=v(t)^2$ and subsequently computing the dynamics of the new variable, i.e., specifying $\dot{z}(t).$ The technique goes back to , where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in , introducing $z$ as an addictional variable yields a quadratic-bilinear control of dimension $n = 3\cdot k$, where $k$ denotes the number of discretization nodes for each PDE, with state vector $x = [v,w,z]^T.$ The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This however opens up the possibility to reduce the system by the generalized moment-matching approach from , see also for more details on the implementation.

Data

In , 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 $n=1024$. All matrices of the quadratic-bilinear formulation discretized with $k=512$ are in the Matrix Market format. The matrix name is used as an extension of the matrix file and can be found at:

FitzNagumo.tar.gz.

For the input function, we have $u(t)=[i_0(t),1]$. For more information on the discretization details, see .

Contact

Tobias Breiten