Interaction between two cells based on a recurrent neural network using the impedance concept

Abstract

In this paper the dynamic behavior of the cells of the immune system is studied through the use of a mathematical and physical model developed using artificial neural networks. Simulations were obtained by solving the equation for the appropriate neuron models, in particular a two cell module. By calculating the equilibrium point of the system and taking into account the bioimpedance of the linearized model, we can determine whether the system is stable or not and whether a self immune disease is developing.

Keywords

Cells recurrent neural network impedance

1. Introduction

The neural networks [1, 2] are very important tools in the field of artificial intelligence. Recently the analogies between the brain and the immune system [3] were exploited to build artificial immune systems and immune algorithms [4, 5, 6] with applications in computer science.

Lymphocyte is a type of white blood cell. The most abundant lymphocytes are the B lymphocytes (B-cells) and the T lymphocytes (T-cells). The B-cells are produced in bone marrow which is a tissue inside the bones. Also the precursors of T-cells produced in the bone marrow and after they mature in the thymus.

The thymus is a gland of the immune system located in the chest between the breast bone and the heart. It has two lobes, the left and the right each being composed of a number of lobules. Each lobule consists of the outer cortex and the inner medulla. The former has a very large number of developing T-cells and a smaller number of associated epithelial cells. T-cells migrate to the medulla in order to mature and to learn how to recognize like-cells from foreign-cells so that inappropriate immune responses are prevented. Immune diseases are caused by different factors such as a lack of the production of enough T-cells because of the improper development of the thymus gland or an excessive immune response or even an autoimmune attack.

The developed models are based on a neural network of a two-neuron module which can be constructed by non linear circuits. By changing the values of the synaptic weights and the activation functions, we can simulate the dynamic behavior of the cells both when the immune system is functioning normally and when it is affected by a disease. With the use of the appropriate computational software, namely Maxima [3] and Maple [4], the model equations can be solved analytically. A related paper has been published before by one of the authors using a three Hopfield neural network [7].

The following Fig. 1 represents a two-neuron recurrent Hopfield neural network with feedback.

Figure 1.

Two neuron network.

It is a network whose purpose is the recognition of the self-antigens in such a way that the immune system does not attack the cells of the body.

2. Methods

The activity of a recurrent two-neuron module at time n, according to Pasemann and Stollenwerk [2], is given by the vector

$\displaystyle x_{n}=(x_{n},y_{n})^{T}$

The discrete dynamic system used to model the neuromodule is given by

$\displaystyle x_{n+1}=\theta_{1}+w_{11}\sigma(x_{n})+w_{12}\sigma(y_{n})$ $\displaystyle y_{n+1}=\theta_{2}+w_{21}\sigma(x_{n})$

where $\sigma$ is the sigmoidal transfer function defining the output of a neuron

$\displaystyle\sigma(x)=\frac{1}{1+e^{-x}}$

$\theta_{1}$ and $\theta_{2}$ are the neuron biases; $w_{ij}$ are weights; the index i indicates the neuron destination for that weight and the index $j$ represents the source of the signal fed to the neuron.

For a two cell module we can calculate the following:

$\displaystyle x_{n+1}=b_{1}+\frac{w_{1,1}}{1+e^{-x_{n}}}+\frac{w_{1,2}}{1+e^{-% y_{n}}}$ (1) $\displaystyle y_{n+1}=b_{2}+\frac{w_{2,1}}{1+e^{-x_{n}}}+\frac{w_{2,2}}{1+e^{-% y_{n}}}$ (2)

Where $x$ and $y$ are the output voltages, $b$ the external voltages and $w$ inductances and mutual inductances of the cells.

Pasemann and Stollenwerk [1] considered the model with the following parameter values:

$\displaystyle b_{1}=-2,b_{2}=3,w_{1,1}=-20,w_{1,2}=6,w_{2,1}=-6,w_{2,2}=0.$ (3)

Equations (1) and (2) become:

$\displaystyle x_{n+1}=-2-\frac{20}{1+e^{-x_{n}}}+\frac{6}{1+e^{-y_{n}}},y_{n+1% }=3-\frac{6}{1+e^{-x_{n}}}.$ (4)

3. Results

When $x_{n+1}=x_{n}$ and $y_{n+1}=y_{n}$ (equilibrium point) then Eq. (4) becomes:

$\displaystyle x=-2-\frac{20}{1+e^{-x}}+\frac{6}{1+e^{-y}},y=3-\frac{6}{1+e^{-x% }}.$ (5)

The solution is:

$\displaystyle x=-1.28037,y=1.69508.$ (6)

If Eq. (5) is written as a vector with the right hand sides then it becomes:

$\displaystyle\left[{-2-\frac{20}{1+e^{-x}}+\frac{6}{1+e^{-y}},3-\frac{6}{1+e^{% -x}}}\right].$ (7)

The Jacobian of Eq. (7) is:

$\displaystyle A=\left[\begin{array}[]{cc}{-\frac{20e^{-x}}{\left({1+e^{-x}}% \right)^{2}}}&{\frac{6e^{-y}}{\left({1+e^{-y}}\right)^{2}}}\\ {-\frac{6e^{-x}}{\left({1+e^{-x}}\right)^{2}}}&0\\ \end{array}\right]$ (8)

If we take into account Eq. (6), then Eq. (8) becomes:

$\displaystyle B=\left[\begin{array}[]{cc}{-\frac{20e^{1.28037}}{\left({1+e^{1.% 28037}}\right)^{2}}}&{\frac{6e^{-1.69508}}{\left({1+e^{-1.69508}}\right)^{2}}}% \\ {-\frac{6e^{1.28037}}{\left({1+e^{1.28037}}\right)^{2}}}&0\\ \end{array}\right]$ (9)

In the case of a non-linear component, the derivative of the voltage (output) with respect to the current (input) is the equivalent resistance. In this case, Eq. (9) is the equivalent bioimpedance [5] for the linearized model.

The eigenvalues (in sec ${}^{-1}$ ) of the impedance matrix Eq. (9) are:

$\displaystyle[-3.14872,-0.25499].$ (10)

Because at least one of the absolute values in Eq. (10) is greater than 1, according to the stability theorem for discrete systems represented by difference equations, the system is unstable.

Subsequent equations are the same as higher memory states. Equation (11) represents the equation that determines the equilibrium point of order 2 (excited memory state), Eq. (6) being the fundamental memory state.

$\displaystyle\left[{x_{n+2}=-2-\frac{20}{1+e^{-x_{n+1}}}+\frac{6}{1+e^{-y_{n+1% }}},y_{n+2}=3-\frac{6}{1+e^{-x_{n+1}}}}\right]$ (11)

The solution is:

$\displaystyle x=-1.28037,y=1.69508$ (12)

The Jacobian of Eq. (13) is

$\displaystyle B=\left[\begin{array}[]{ccc}{-\frac{20e^{14.18518916}}{\left({1+% e^{14.18518}}\right)^{2}}}&{\frac{6e^{2.37301}}{\left({1+e^{2.37301}}\right)^{% 2}}}&{\frac{5e^{-0.98117}}{\left({1+e^{-0.98117}}\right)^{2}}}\\ {-\frac{6e^{14.18518}}{\left({1+e^{14.18518}}\right)^{2}}}&0&0\\ 0&0&{-\frac{10e^{-0.98117}}{\left({1+e^{-0.98117}}\right)^{2}}}\\ \end{array}\right]$ (13)

The eigenvalues (in sec ${}^{-1}$ ) of the impedance matrix of Eq. (13) are

$\displaystyle[-3.14872,-0.25499]$ (14)

Using the following programs we get Figs 2 and 3 respectively.

Figure 2.

Chaotic attractor for a minimal chaotic neuromodule.

Program 1:

x: $=$ array(0..100000):y: $=$ array(0..100000):

b1: $=$ $-$ 2:b2: $=$ 3:w11: $=$ $-$ 20:w21: $=$ $-$ 6:w12: $=$ 6:imax: $=$ 10000:

x[0]: $=$ 1:y[0]: $=$ 0.2:

for i from 0 to imax do

x[i $+$ 1]: $=$ evalf(b1 $+$ w11/(1 $+$ exp( $-$ x[i])) $+$ w12/(1 $+$ exp( $-$ y[i]))):

y[i $+$ 1]: $=$ evalf(b2 $+$ w21/(1 $+$ exp( $-$ x[i]))):

end do:

with(plots):

points: $=$ [[x[n],y[n]]$n $=$ 50..imax]:

pointplot(points,style $=$ point,symbol $=$ point,color $=$ blue,axes $=$ BOXED,

font $=$ [TIMES,ROMAN,15]);

Figures 2 and 3 show the dynamic behaviour of the cells of the immune system, particularly the immune memories and the bifurcations from normal behaviour (Fig. 2) to abnormal behaviour (Fig. 3).

Figure 3.

Bifurcation diagram for a bistable neuromodule which displays quasiperiodic and bistable behaviors.

The horizontal axis shows the voltage gradient between the inside and outside of a cell and the vertical axis shows the voltage of another cell in a system of two with feedback.

Program 2:

start: $=$ 7:Max: $=$ 7:b2: $=$ 3:b1: $=$ 2:w12: $=$ $-$ 4:w21: $=$ 5:

halfN: $=$ 9999:N1: $=$ 1 $+$ halfN:itermax: $=$ 2*halfN $+$ 1:

x[0]: $=$ $-$ 3:y[0]: $=$ $-$ 2:

for n from 0 to halfN do

w11: $=$ start $-$ n*Max/halfN:

x[n $+$ 1]: $=$ evalf(b1 $+$ w11*tanh(x[n]) $+$ w12*tanh(0.3*y[n])):

y[n $+$ 1]: $=$ evalf(b2 $+$ w21*tanh(x[n])):

end do:

with(plots):

points: $=$ [[start $-$ j*Max/N1,x[j]]$j $=$ 0..halfN]:

P1: $=$ pointplot(points,style $=$ point,symbol $=$ point,color $=$ blue,axes $=$ BOXED,

font $=$ [TIMES,ROMAN,15]);

for n from N1 to itermax do

w11: $=$ (n $-$ N1)*Max/halfN:

x[n $+$ 1]: $=$ evalf(b1 $+$ w11*tanh(x[n]) $+$ w12*tanh(0.3*y[n])):

y[n $+$ 1]: $=$ evalf(b2 $+$ w21*tanh(x[n])):

end do:

points: $=$ [[start $+$ (j $-$ N1)*Max/N1,x[N1 $+$ j]]$j $=$ 0..halfN]:

P2: $=$ pointplot(points,style $=$ point,symbol $=$ point,color $=$ blue,axes $=$ BOXED,

font $=$ [TIMES,ROMAN,15]);

display({P1,P2},labels $=$ [’w11’,’x[n]’]);

Figure 3 shows how the state of a cell changes when the connectivity between the cells is altered namely that the memories are lost and the immune system will attack the body and create self-immune disease.

With the use of the next program we get a numerical solution for a system of two normal cells and the equivalent Fig. 4 which represents a stable immune memory and a normal immune system. The immune system keeps the memory of the self-antigens and does not attack the body.

Figure 4.

System of two normal cells.

Program 3:

restart:

x: $=$ array(0..100000):y: $=$ array(0..100000):

imax: $=$ 15000:

x[0]: $=$ 1:y[0]: $=$ 0.2:

for i from 0 to imax do

x[i $+$ 1]: $=$ evalf(2 $+$ 3.5*tanh(x[i]) $-$ 4*tanh(0.3*y[i])):

y[i $+$ 1]: $=$ evalf(3 $+$ 5*tanh(x[i])):end do:

with(plots):

points: $=$ [[x[n],y[n]]$n $=$ 50..imax]:

pointplot(points,style $=$ point,symbol $=$ point,color $=$ blue,axes $=$ BOXED,

font $=$ [TIMES,ROMAN,15]);

4. Conclusions

The goal of this paper was to develop a model based on a two-neuron recurrent Hopfield neural network with feedback to simulate the interactions between the two cells considering that the bioimpedance of the artificial neuronal network state variable is represented by the transfer functions from antigens to immune response. By calculating the equilibrium point of the system and taking into account the bioimpedance of the linearized model we can find out if the system is stable or not.

Footnotes

Acknowledgments

The authors thank Ms Kate Somerscales for proofreading the manuscript.

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