Radu Constantinescu , Carmen Ionescu, Mihai Stoicescu Radu - - PowerPoint PPT Presentation

Radu Constantinescu , Carmen Ionescu, Mihai Stoicescu Radu Constantinescu , Carmen Ionescu, Mihai Stoicescu D f f Ph Ph i U i i f f C i Department epartment of Ph f Phys ysics, cs, U Univers versity ty of C f Cra raiova

• va

1. Neuron models: from real world to mathematical and technical models. 2 Membrane phenomena in ne ral cells 2. Membrane phenomena in neural cells. 3. Cable equation of the axon. 4. Neural networks. Hopfield type models.

Models for neurons

From real world to mathematical and technical models

y models? y models?

• Modeling – the natural mental action in understanding the world
• The model should be realistic
• It has to capture the biological complexity but to allow further

stigations. dels may describe an object or a phenomenon as:

• Structure (mechanical model)

Structure (mechanical model)

• Function (electronic or mathematical or computer model)
• Evolution (transport through membrane or along axon, synaptic

transmission) main neuron models are expressed in terms of

• Mathematical equations (Hodgkin-Huxley equations)
• An imaginary construction following the laws of physics (Eccles model)

An imaginary construction following the laws of physics (Eccles model) El t i d l ti l i l th i i l h

Membrane phenomena in neural cells

ctric conduction throught membrane tric conduction throught membrane

p

W Q(Φ Φ ) F(

)

he energy spent as work of electrical force to move an electric charge along electric circuit is given by:

Φ0 ; Φi = extra and intracellular potentials

(1) We = Q(Φ0 - Φi) = zF(Φ0 - Φi)

z

= valence of the ions

F

= Faraday's constant [9.649 × 104 C/mol]

nsport

• rt phenomena throu

henomena through membrane h membrane

(1)

p p p p g

According to Ohm's law, current density and electric field are related by

where σ is the conductivity of the medium

(2) r a system with many types of ions:



k

z

uk = ionic mobility [cm²/(V·s)]

where:

(3)

  

k k k

c u 

zk = valence of the ion

(3)

• rding to Fick’s Law:

where: T = absolute temperature [K] R = gas constant [8.314 J/(mol·K)]

st – Plank Equation for neuron:

JkD = ionic flux (due to diffusion) [mol/(cm²·s)] Dk = Fick's constant (diffusion constant) [cm²/s] ck = ion concentration [mol/cm³]

c flux

ci,k = intracellular concentration of the kth ion co,k = extracellular concentration of the kth ion

tric current density

brium condition for current density:

F

 

 k

J

     RT F z c c

k k k

ivalent electric circuit for neural cell ivalent electric circuit for neural cell

neuron membrane behaves as a capacitor with many conducting equivalent electric circuit is presented below and the Kirchhoff laws neuron membrane behaves as a capacitor with many conducting nels joining its two sides, the extracellular and the cytoplasmic ones.

 



j

=>

1





 n k k kI



1 k

 

 

E j  1

 

 



m k k n k k k k

U I R

1 1



I I I I

ith the concrete notations:

Ic +INa +IK + IL =0

m L L K K Na Na

I V V g V V g V V g dt dV         ) ( ) ( ) (

Cable equation for the axon q

Suppositions: Suppositions: the voltage depends on the point where is calculated the flow travel with constant velocity and the pulse maintains its

• riginal form during the propagation
• riginal form during the propagation.

) , ( t x U U  Conclusion: the dynamics obeys the wave equation and the Conclusion: the dynamics obeys the wave equation and the

• The propagation equation has the form:

U U

2 2

1   

The propagation equation has the form:

t x

2 2 2

   

• Suppose the variation of the potential along the axon satisfies the relation:

0r

I r I x U

i i

   

With d i ti f th i l ti t With new derivation of the previous relation we get:

x I r x I r x U

i i

       

2 2

The conservation of total charge imposes

m i

x I x I       

• Substituting this last equation in the wave equation, we obtain

2 2

1 U U   1 ) ( U U r ri      

Neural networks. Hopfield type models p yp

• posed in 1982 by John Hopfield to explain the functioning of the
• posed in 1982 by John Hopfield to explain the functioning of the

mory as an asynchronous neural network.

• nsists in n totally coupled units, that is, each unit is connected to all

er units except itself and change information recursively between m. he individual units are randomly updated preserving their individual he individual units are randomly updated, preserving their individual tes in the interval between two updates. uppositions:

• synchronization requirements (not a universal time is needed)

he neurons are not updated simultaneously ymmetric transfer of information among units ymmetric transfer of information among units

The starting point is the Hodgkin-Huxley model and the Kirchhoff law:

dV

Suppositions:

m L L K K Na Na m

I V V g V V g V V g dt dV C         ) ( ) ( ) (

pp

• the potential is constant along the axon.
• in the same unit area, there is a delay between the voltage U on the

esistive channels and the capacitive potential V. The cable equation of the axon can be written as:

) ( ) , , (   

m m

t I t U V f dt dV C ) , , ( ); , , (     

V U

t U V g t U V g dt dU dt

The previous system gives an expression of the voltage V in terms of U.

Mathematical form of the binary model Mathematical form of the binary model

Experiments shown that there are two branches: Experiments shown that there are two branches:

• a mostly liner branch
• a second nonlinear one

The two branches can be separated by using a binary variable S=+1

• r -1.

From the previous consideration we conclude that the functioning of

• ne neuron can be described in good agreement with the real neuron

behavior by the equation:

) ( ) 1 ( ) ( t I S c t U b a dU

m

    

For many neurons, the Hopfield model proposed the equations:

) ( ) 1 ( ) ( t I S c t U b a dt

m



n

n k t u G w u b t u 1 ) ( ) ( ) (       

merical simulations for the erical simulations for the action potential action potential

ing back to the cable equation and passing to the “wave” coordinate: previous equation becomes:

c x t   

re: 1 



• BU
• AU

U

' ''

re: studied numerically the behavior of the voltage along the wave, and the d i h fi b l

2 2

1 ;      B A ts are presented in the figure below:

Conclusions: Conclusions:

The propagation of neural flow and across neurons is a not yet The propagation of neural flow and across neurons is a not yet cidated problem. cidated problem. There are many models from very complex as Hodgkin There are many models from very complex as Hodgkin-

• Huxley,

Huxley, tinuing with integrate or fire models, till very simple ones as binary tinuing with integrate or fire models, till very simple ones as binary pfield pfield-

• type models.

type models. The reduction of the complexity of a model has to allow detailed The reduction of the complexity of a model has to allow detailed dies on network characteristics but not to spoil its biological content. dies on network characteristics but not to spoil its biological content. Using Hopfield model, we described the flow in terms of a simple Using Hopfield model, we described the flow in terms of a simple ameter A: it expands for small values of ameter A: it expands for small values of A and strongly damp for and strongly damp for

Thank you ! Thank you !