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 dynamics of the system is governed by the following coupled nonlinear PDEs:

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

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

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. Fig. 1 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 $B \in \mathbb R^{n\times 2}$, and the input function is $u(t)=[i_0(t),1]$. 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 model reduction schemes developed for quadratic-bilinear systems such as balanced truncation , or interpolation-based approaches, e.g., given in and .

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 $2\cdot k=1024$. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension $3\cdot k=1536$, and all matrices of the quadratic-bilinear formulation discretized 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.

The archive contains the matrices $E$, $A$, $B$, $H$ and $N_1$; The matrix $N_2$ is a zero matrix of appropiate size.

For more information on the discretization details, see .

In addition, one can have an output

$y(t) = Cx(t),$

where the matrix $C$ is a $2\times N$ matrix such that $C(1,1) = 1, C(2,1+k) = 1$ and the other elements are zero. This output $y(t)$ corresponds to the left boundary of the limit cycles. For more information on the output, see .

Dimensions

System structure:

$\begin{align} E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{i=1}^2N_i x(t) u_i(t) + B u(t) \\ y(t) &= Cx(t) \end{align}$

System dimensions:

$E \in \mathbb{R}^{1\,536 \times 1\,536}$, $A \in \mathbb{R}^{1\,536 \times 1\,536}$, $H \in \mathbb{R}^{1\,536 \times 2\,359\,296}$, $N_1 \in \mathbb{R}^{1\,536 \times 1\,536}$, $N_2 \in \mathbb{R}^{1\,536 \times 1\,536}$, $B \in \mathbb{R}^{1\,536 \times 2}$, $C \in \mathbb{R}^{2 \times 1\,536}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, FitzHugh-Nagumo System. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/FitzHugh-Nagumo_System

@MISC{morwiki_modFHN,
  author =       {{The MORwiki Community}},
  title =        {FitzHugh-Nagumo System},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/FitzHugh-Nagumo_System},
  year =         2018
}

• For the background on the benchmark: morChaS10 (BibTeX)

Contact

Tobias Breiten