Abstract
Electrical activities of excitable cells produce diverse spiking-bursting patterns. The dynamics of the neuronal responses can be changed due to the variations of ionic concentrations between outside and inside the cell membrane. We investigate such type of spiking-bursting patterns under the effect of an electromagnetic induction on an excitable neuron model. The effect of electromagnetic induction across the membrane potential can be considered to analyze the collective behavior for signal processing. The paper addresses the issue of the electromagnetic flow on a modified Hindmarsh–Rose model (H–R) which preserves biophysical neurocomputational properties of a class of neuron models. The different types of firing activities such as square wave bursting, chattering, fast spiking, periodic spiking, mixed-mode oscillations etc. can be observed using different injected current stimulus. The improved version of the model includes more parameter sets and the multiple electrical activities are exhibited in different parameter regimes. We perform the bifurcation analysis analytically and numerically with respect to the key parameters which reveals the properties of the fast-slow system for neuronal responses. The firing activities can be suppressed/enhanced using the different external stimulus current and by allowing a noise induced current. To study the electrical activities of neural computation, the improved neuron model is suitable for further investigation.
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Acknowledgements
This work is supported by the Council of Scientific and Industrial Research (CSIR), Govt. of India under Grant No. 25(0277)/17/EMR-II) to the author (R. K. Upadhyay).
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Appendices
Appendix A
The equilibrium points are derived from the following system of equations
In the following, we consider the nature of roots of Eq. (4a) to obtain the values of \(u^*\). Eq. (4a) can be written as \(f({u^ * }) = {p_0}{u^{ * 3}} + {p_1}{u^{ * 2}} + {p_2}{u^ * } + {p_3} = 0\), where \({p_0} = ({a_1}s - 3\beta ({k_1}/k_2^2)),\)\({p_1} = - (s + 1)\), \({p_2} = - ({b_1}s({a_2}/k) + {k_1}\alpha )\), \({p_3} = - ({b_1}{b_2}/k)\) with the condition \({a_1}sk_2^2 \ne 3\beta {k_1}\) at the preassigned parameter set. The number and types of the roots are determined using the discriminant of the cubic equation. The general solution of the equation involves the following expressions
\({\Delta _0} = p_1^2 - 3{p_0}{p_2},\)\({\Delta _1} = 2p_1^3 - 9{p_0}{p_1}{p_2} + 27p_0^2{p_3},\) and \({D_1} = \root 3 \of {{\left( {{\Delta _1} \pm \sqrt{\Delta _1^2 - 4\Delta _0^3} } \right) /2}}\).
The general rule to find all the roots of the cubic equation is of the following form \(u_n^ * = - (1/3{p_0})({p_1} + {\xi ^n}{D_1} + {\Delta _0}/({\xi ^n}{D_1}))\)\((n=0, 1, 2)\), where \(\xi = - (1/2) + (1/2)\sqrt{3}i\) (which denotes a cube root of unity). Now, we can easily obtain the corresponding roots of the equation.
Appendix B
The variational matrix is derived from the system 1 as follows
The coefficients of the characteristic Eq. 2 are as follows
-
\({m_0}=1\), \({m_1} = {k_2} + k\varepsilon + 1 - 3{a_1}s{u^{ * 2}} + 2{u^ * }s + {k_1}\alpha + 3\beta {k_1}{w^{ * 2}}\),
-
\({m_2} = k{k_2}\varepsilon + {k_2} + ({k_2} + k\varepsilon + 1)( - 3{a_1}s{u^{ * 2}} + 2s{u^ * } + {k_1}\alpha + 3\beta {k_1}{w^{ * 2}}) + k\varepsilon + 2{u^ * } + {a_2}{b_1}\varepsilon s + 6\beta {k_1}{u^ * }{w^ * }\),
-
\({m_3} = k{k_2}\varepsilon + (k{k_2}\varepsilon + k\varepsilon + {k_2})( - 3{a_1}s{u^{ * 2}} + 2s{u^ * } + {k_1}\alpha + 3\beta {k_1}{w^{ * 2}}) + 2{u^ * }({k_2} + k\varepsilon )\)\( + {a_2}{b_1}\varepsilon s({k_2} + 1) + 6\beta {k_1}{u^ * }{w^ * }(k\varepsilon + 1)\),
-
\({m_4} = k{k_2}\varepsilon ( - 3{a_1}s{u^{ * 2}} + 2s{u^ * } + {k_1}\alpha + 3\beta {k_1}{w^{ * 2}}) + 2k{k_2}\varepsilon {u^ * } + {a_2}{b_1}{k_2}\varepsilon s + 6\beta k{k_1}\varepsilon {u^ * }{w^ * }\).
Appendix C
The complex roots are \({\lambda _1},\,\,{\lambda _2} = - \frac{{{m_1}}}{{4{m_0}}} + S \pm \frac{1}{2}\sqrt{ - 4{S^2} - 2P - Q/S}\), where \(P = (8{m_0}{m_2} - 3m_1^2)/8m_0^2,\,\,Q = (m_1^3 - 4{m_0}{m_1}{m_2} + 8m_0^2{m_3})/8m_0^3\), \(S = 0.5\sqrt{ - (2/3)P + (1/3{m_0})(R + {\Delta _3}/R)} \), \(R = \root 3 \of {{0.5({\Delta _2} + \sqrt{\Delta _2^2 - 4\Delta _3^3} )}}\), \({\Delta _2} = 2m_2^3 - 9{m_1}{m_2}{m_3} + 27m_1^2{m_4} + 27{m_0}m_3^2 - 72{m_0}{m_2}{m_4}\) and \({\Delta _3} = m_2^2 - 3{m_1}{m_3} + 12{m_0}{m_4}\).
Genericity condition is as follows
where \(f_{ij}^k = {\left. {\frac{{{\partial ^2}{f^k}}}{{\partial {y_i}\partial {y_j}}}} \right| _{(0,{b_{2(crt)}})}}\) and \(f_{ijl}^k = {\left. {\frac{{{\partial ^3}{f^k}}}{{\partial {y_i}\partial {y_j}\partial {y_l}}}} \right| _{(0,{b_{2(crt)}})}}\)\((i, j, k=1, 2, 3, 4)\). Now, and M is a nonsingular matrix such that \({M^{ - 1}}AM = U\) at \({b_2}={b_{2(crt)}}\), \(y = {({y_1},\,\,{y_2},\,\,{y_3},\,\,{y_4})^T}\). F is the nonlinear functions and it becomes
\(F = {(s{a_1}{u^3} + 3s{a_1}{u^2}{u^ * } - s{u^2} - 3{k_1}\beta {w^2}(u + {u^ * }) - 6{k_1}\beta uw{w^ * },\,\,\,{u^2},\,\,0,\,\,0)^T}.\)
The matrix U becomes
A is the Jacobian matrix whose elements are \({a_{11}} \approx 0.9511832408,{a_{12}} = - 1,{a_{13}} = - 1,{a_{14}} \approx - 0.1022020445\), \({a_{21}} \approx 2.06359359,{a_{22}} = - 1,{a_{23}} = 0,{a_{24}} = 0, {a_{31}} = 0.0182,{a_{32}} = 0,{a_{33}} = - 0.014,{a_{34}} = 0, {a_{41}} = 1,{a_{42}} = 0,{a_{43}} = 0,{a_{44}} = - 0.5\) around the fixed point \(E^*\) with the critical value of \(b_2\) at parameter Set I. Suppose \(v_1, v_2, v_3\) be the eigenvectors corresponding to the eigenvalues \({\lambda _1},\,\,{\lambda _2} = \pm i\beta ,{\lambda _3} = \mu ,\,{\lambda _4} = \psi \), then consider the nonsingular matrix \(M=\) col \((Re({v_1}),Im({v_1}),{v_2},{v_3})\) and the elements are derived as \({m_{11}} \approx 0.35329475,{m_{12}} \approx 0.395077699,{m_{13}} \approx - 0.0345995049, {m_{14}} \approx - 0.2907838,{m_{21}} \approx 0.72919971,{m_{22}} = 0,{m_{23}} \approx - 0.1535589, {m_{24}} \approx - 0.616957,{m_{31}} \approx 0.006501239,{m_{32}} \approx - 0.00567079, {m_{33}} \approx 0.00120857,{m_{34}} \approx 0.39528619,{m_{41}} \approx 0.4122414,{m_{42}} \approx - 0.131707,{m_{43}} \approx 0.9875328,{m_{44}} \approx - 0.6152702\).
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Mondal, A., Upadhyay, R.K., Ma, J. et al. Bifurcation analysis and diverse firing activities of a modified excitable neuron model. Cogn Neurodyn 13, 393–407 (2019). https://doi.org/10.1007/s11571-019-09526-z
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DOI: https://doi.org/10.1007/s11571-019-09526-z