Example: Neuron model

A model of a single neuron cell is given by the differential equation:

$\displaystyle \begin{pmatrix}\dot{x}\\ \dot{y}\\ \dot{z} \end{pmatrix}= \begin{...
...\alpha \mbox{Inp}\\ -c-dx^2-\varphi_3 x-\beta y\\ r [s(x+x_0)-z] \end{pmatrix}.$ (7.4)

In this model, the state $ x(t)$ represents the electric potential the neuron produces. Parameters $ a, b, c, d, r, s, x_0, g_y, g_z, \alpha, \beta, \varphi_1, \varphi_2$ and $ \varphi_3$ are constants. Inp is the input of the model.

Suppose we want to simulate this system for 1000 ms with initial conditions $ x_0=0$ , $ y=0$ and $ z=0$ . Hereto, a Simulink scheme is created. To compute the signals $ x^2$ and $ x^3$ , a new block is used: Fcn. In the Simulink library, this block is found under "User-Defined Functions". In the parameters of this block, one can enter the expression this function calculates, where u is the input of the block.

In addition, a somewhat different appearance is chosen for the Sum block. This block has the option "Icon shape". The default value "round" is used in the previous sections. Now, we use "Rectangular". To add or substract more signals, the option "List of signs" can be changed. For example, adding the first two signals and substracting a third one is achieved by the list of signs "++-".

The neuron model is implemented in Simulink scheme in Figure 7.1:

Figure 7.1: Simulink scheme of neuron model
\begin{figure}\centering
\epsfig{file=ch7_neuron1.eps,width=800px}
\end{figure}

When simulating this model, the variables $ a, b, c, d,$ $ r, s, x_0,$ $ gy$ , $ gz$ , $ Alpha$ , $ Beta$ , $ Phi1$ , $ Phi2$ , $ Phi3$ should be defined in Matlab his Workspace. For certain parameters, the $ x$ -trajectory depicted in Figure 7.2 is obtained for the constant input Inp$ =3.3$ :

Figure 7.2: Time trajectory of neuron model
\begin{figure}\centering
\epsfig{file=ch7_hrtraj,width=.7\linewidth}
\end{figure}

Since this model is stiff, the solver type ode15s is used. The Simulink file of this example can be downloaded here.


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Esteur 2010-03-22