Study of neuron models

by Sorin Mitran

Abstract

This report describes the classic Hodgkin-Huxley model of neuron electrochemical activity that leads to synapses, extended to consideration of unidimensional propagation along a sequence of neurons.

1Introduction

Multicellular biological organisms with differentiated cells need to establish information transfer between various specialized cells. We consider an extension of the classical Hodgkin-Huxley electrochemical neuron model [?] to series of neurons.

1.1Neuron physiology

1.1.1Neuron structure

Figure 1. Basic structure of a neuron.

As shown in Fig. 1, a neuron is a specialized cell with an elongated extension called an axon and several shorter extensions called dendrites.

1.1.2Electrochemistry of the neuron

\(\circ\)Neuron structure maintains a resting polarization voltage across an axon membrane of \(V_r = - 70\) mV.

1.2Hodgkin-Huxley model

1.2.1Equivalent electrical circuit

Figure 2.

1.2.2Ion transport across neuron membrane

1.3Propagation of electrical impulses along neuron axon

2Methods

2.1Change of ion concentration in axon cross-section

Assuming constant ion concentrations in an axon cross-section the rate of change of K\(^+\), Na\(^+\), Cl\(^-\) concentrations is given by the ODE system

\(\circ\)

\(\displaystyle \frac{\mathrm{d} m}{\mathrm{d} t} = \alpha_m (V) (1 - m) - \beta_m (V) m\)

\(\circ\)

\(\displaystyle \frac{\mathrm{d} n}{\mathrm{d} t} = \alpha_n (V) (1 - n) - \beta_n (V) n\)

\(\circ\)

\(\displaystyle \frac{\mathrm{d} h}{\mathrm{d} t} = \alpha_h (V) (1 - h) - \beta_h (V) h\)

The equations express a two time scale concentration evolution between values of 0 and 1 with time scales given by \(\alpha_i, \beta_i\) respectively (Fig. )

2.2Membrane voltage

2.3Driving force

Ions are transported across the axon membrane due to a voltage difference. Transport occurs across ion channels, both passively and actively. The overall rate of change of voltage is given by

\(\bullet\)

\(\displaystyle C_m \frac{\mathrm{d} V}{\mathrm{d} t} = \bar{g}_{\operatorname{Na}} m^3 h (V - E_{\operatorname{Na}}) + \bar{g}_{\text{K}} n^4 (V - E_{\text{K}}) + g_{\ell} (V - E_{\ell}) + P (t)\) 

In[20]:= 
iNa = gNA (m[t])^3 h[t] (V[t] - ENa);
iK = gK (n[t])^4 (V[t] - EK);
iCl = gl (V[t] - El);
 

In[23]:= 
ODE4 = Cm V'[t] == iNa + iK + iCL + P[t];
 

In[24]:= 

              

            

          

        

      

    

    

      \(\displaystyle \)
    

    

      

        \(\circ\)The resting voltages for the individual ions are
        \(E_{\operatorname{Na}} = 55\) mV, \(E_K = - 82\) mV, \(E_{\ell} = -
        59\) mV
      

      
    

    

      

        \(\circ\)Ion conductance constants are
        \(\)\(\bar{g}_{\operatorname{Na}} = 70.7\) (m-mhos/cm\(^2\)), 
        \(\)\(\bar{g}_K = 24.34\) (m-mhos/cm\(^2\)), \(g_{\ell} = 0.3\)
        (m-mhos/cm\(^2\)).
      

      
    

    

      

        \(\circ\)The membrane capacitance is \(C_m = 0.001\) (F/cm\(^2\))
      

      
    

    

      

        

          

            
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ODEsys = {ODE1,ODE2,ODE3,ODE4};

          

        

      

    

    

      

        

          

            

              
            

          

        

      

    

    

      

        

          

            
In[16]:= 

            
IC = {m[0]==1,n[0]==0.5,h[0]==0.25,V[0]=-70};

          

        

      

    

    

      

        

          

            

              
            

          

        

      

    

    

      

        

          

            
In[21]:= 

            
am[V_]=V; an[V_]=V; al[V_]=V; bm[V_]=1; bn[V_]=1; bl[V_]=V;

          

        

      

    

    

      

        

          

            

              
            

          

        

      

    

    

      

        

          

            
In[18]:= 

            
gNA=70.7; gK=24.34; gl=0.5;

          

        

      

    

    

      

        

          

            

              
            

          

        

      

    

    

      

        

          

            
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P[t_]=0;

          

        

      

    

    

      

        

          

            

              
            

          

        

      

    

    

      

        

          

            
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sol=NDSolve[Flatten[ODEsys,IC],{m[t],n[t],l[t],V[t]},{t,0,1}];

          

        

      

    

    

      

        

          

            

              Flatten::flpi: Levels to be flattened together in
              m[0] == 1, n[0] == 0.5, h[0] == 0.25, -70 should be lists of
              positive integers. 
            

            

              NDSolve::deqn: Equation or list of equations
              expected instead of Flatten[m'[t] == -m[t] + (1 - m[t]) V[t],
              n'[t] == -n[t] + (1 - n[t]) V[t], h'[t] == ah[<<1>>]
              <<1>> + <<1>>, 4 3 Cm V'[t] == iCL +
              n[t] (-EK + V[t]) + h[t] m[t] (-ENa + V[t]), <<4>>]
              in the first argument Flatten[m'[t] == -m[t] + (1 - m[t]) V[t],
              n'[t] == -n[t] + (1 - n[t]) V[t], h'[t] == ah[<<1>>]
              <<1>> + <<1>>, 4 3 Cm V'[t] == iCL +
              n[t] (-EK + V[t]) + h[t] m[t] (-ENa + V[t]),
              <<4>>]. 
            

          

        

      

    

    

      

        

          

            
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3Results

    
4Conclusions

    
Bibliography