# Study on Bifurcation of H-H Parameters And Its Variants Contribution To Neurology

**Taslima Ahmed*, Jiten Ch. Dutta**

Department of ECE, Tezpur University (a Central University), Napaam Post, Tezpur, Assam, India

*Corresponding Author

Taslima Ahmed,

Department of ECE,

Tezpur University (a Central University),

Napaam Post, Tezpur, Assam,India.

**E-mail**: ahmed.taslima@gmail.com

**Article Type**: Research Article

**Received**: July 30, 2014; **Accepted**: July 22, 2014; **Published**: July 24, 2014

**Citation**: Taslima Ahmed, Jiten Ch. Dutta(2014) Study on Bifurcation of H-H Parameters And Its Variants Contribution To Neurology. Int J Comput Neural Eng, 1(2) 6-10. **doi: dx.doi.org/10.19070/2572-7389-140002**

**Copyright**: Taslima Ahmed© 2014. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

Abstract

The important application of bifurcation analysis of a system is estimated by the variables of H-H model. Graphical User Interface (GUI) is essential for proper visualization of results and therefore, here, we are discussing some GUI based bifurcation panels. Based on these panels, we can investigate different abnormal disorders for neurological applications.

2.Introduction

3.Analysis of bifurcation in some models

3.1 GUI Morris- Lecar (ML) model by B.Raesi

3.2 Jiang Wang bifurcation for gl value

3.2 Jiang Wang for non-linear model

4.Conclusion

5. References

Keywords

H-H model; Bifurcation; Neurology.

Introduction

Neuron is the fundamental unit for transmitting signals in nervous system [1-3]. The biological membrane is also played an important role in many of life’s processes [4].Many of these processes are electrical and the different electrical behavior of nerve cells can be measured experimentally. The flow of ions across the membrane is responsible for the production of membrane potential. The mathematical formulation of the function of neuron was given firstly by H-H model [5-8]. The variation of parameters of H-H equations leads to bifurcation which refers to quantitative changes in the solution structure of dynamical systems.

Neuroscience computes different problems based on neural model from the analysis of bifurcation in H-H variables. This analysis progress in modern quantitative biology and biophysics [9].

It is the main significant issue to control bifurcation because many neuronal disorders are due to bifurcation of neuronal systems. In the field of neurology, these disorders present Alzheimer’s disease epilepsy and arrhythmia [10,11]. Bifurcation control has been employed to estimate seizing behavior in the model system of human cortical electrical activity [12].

In this paper, firstly we describe the GUI Morris- Lecar baesd [13] model for global phase portraits. Secondly, we show GUI panel based on the analysis by Jiang Wang [14] that finds the bifurcation would occur when the leakage conductance gl is lower than 0.299406mS/cm2. Thirdly, we again show the analysis by Jiang Wang [15] for the investigation of the synchronization of Fitz- Hugh-Nagumo neural system under external electrical stimulation via the nonlinear control by using GUI. Lastly, we discuss the application of H-H variants for contribution in neuroscience based on these example panels.

Analysis of bifurcation in some models

From this table-1, we discuss different bifurcation analysis as follows:

17],[18-20],19,21,22,23

In this ML model, to solve the equation 1, we first need to create functions m∞(v) and n∞(v). With the help of these functions, we take the different values of different parameters for plotting the various sub panels. Again, with the help of MATLAB library function “ode 23”, we solve all the differential equations2, 3, 4, 5.The solutions are plotted by taking the different values from the table-2. We plot the result of main function capacitive function Cv w. r. t time, n w. r. t time, m w. r. t time respectively. Then the different phase portraits are obtained by plotting the graph by taking three variable Cv, n, m, w. r. t each other as shown in plot (a), plot (b). The equations are shown as:

Where

The HH model can be described using following four equations:

dm/dt= fm(V,m)=αm(V) (1-m)-βm (V)m

dh/dt= fh(V,h)= αh(V) (1-h)-βh(V)h

dn/dt= fn(V,n)= αn(V) (1-n)-βn(V)n

where, αm= 0.08 (V+56)/(1-exp(-(V+56)/6.8)

βm = 0.8exp(-(V+56)/18)

αh= 0.006exp(-(V+41)/14.7)

βh= 1.3 /(1+exp(-(V+41)/7.6)

αn= 0.0088 (V+40)/(1-exp(-(V+40)/7)

βn = 0.037exp(-(V+40)/40)

Where, V=membrane voltage, Iext =injected current, n=activation variable of potassium channel, m=activation variable of sodium channel, h=inactivation variable of sodium channel, Cm=1.9μF/cm2, gna=50 mS/ cm2, gk=22 mS/ cm2, gl=0.4 mS/ cm2, vna=50V, vk=-70V,vl=-81V.

By putting all the parameter’s value, we can find that there is only one bifurcation parameter which is gl..So to analyze the effect the leakage current parameter gl .We did the partial differentiation of above four equations and obtains a Jacobian matrix as shown:

So the coefficients of Jacobian matrix are the solution find the partial differentiation of those equations and putting all values of parameters.

In the nonlinear cable model, the model equation is a coupled differential equation.

We use “ode15” function in MATLAB to solve these equations. After solving the equations the solution is plotted with respect to time having different frequencies as shown in figure-2.

The non linear model equations are given as follows with transmembrane voltage V along the nerve fibre as:

### dX/dt= X(X-1)(1-rY)-Y+I(t)

dY/dt=bX

where

X,Y are membrane voltages, W=recovery variable

vp=peak of action potential

X=V/ vp, Y=W/ vp, r= vp/vT, where, vT=threshold membrane voltage

I(t)=A/ᴡ coswt, A=strength of applied field

W=angular frequency of applied field,

w=2ᴨf, f=frequency

Different observations at different frequency values are concluded with table-3 form the plot figure-3

The performance of the transmembrane potantials X with various frequency f:

(a)1-periodic oscillation at f=25 Hz1MB

(b)quasiperiodic oscillation at f=25.1 Hz1

(c)1-periodic oscillation again at f=35 Hz1

the model equation is given bellow:

dX/dt=X(X-1)(1-rX)-Y+l(t)

dY/dt=dX

Understanding complex neurobiological systems is one of the most difficult challenges in modern science [24]. From these above results and discussion, we can conclude that H-H equation is the foundation of neuroscience as these parameters values are used for computational brain modeling. It removes ambiguity from theories and makes them logically consistent. Use of computer technology enables theories involving with a large number of elements to be investigated. Computational modeling can help to do the right experiment to solve numerically a set of biologically grounded equations describing the voltage-dependent changes. Computer modeling is an essential component of the neuroscientist’s repertoire. Any variation of the H-H parameters can cause bifurcation and this analysis can solve different abnormal disorders by investigating the graphs as shown above. Without H-H model, there is no existence of research in neuroscience as today. So, from these different panels based on H-H equations can solve the problem of investigation of different diseases by researchers also.

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