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Bifurcation analysis and diverse firing activities of a modified excitable neuron model

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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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References

  • Baltanas JP, Casado JM (2002) Noise-induced resonances in the Hindmarsh-Rose neuronal model. Phys Rev E 65(4):041915

    Article  CAS  Google Scholar 

  • Bao B, Jiang T, Xu Q et al (2016) Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn 86(3):1711–1723

    Article  Google Scholar 

  • Bao BC, Liu Z, Xu JP (2010) Steady periodic memristor oscillator with transient chaotic behaviors. Electron Lett 46(3):237–238

    Article  Google Scholar 

  • Barrio R, Shilnikov A (2011) Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model. J Math Neurosci 1(1):6

    Article  PubMed  PubMed Central  Google Scholar 

  • Bekkers JM (2003) Synaptic transmission: functional autapses in the cortex. Curr Biol 13(11):R433–R435

    Article  CAS  PubMed  Google Scholar 

  • Bertram R, Rubin JE (2017) Multi-timescale systems and fast-slow analysis. Math Biosci 287:105–121

    Article  PubMed  Google Scholar 

  • Chik DTW, Wang Y, Wang ZD (2001) Stochastic resonance in a Hodgkin-Huxley neuron in the absence of external noise. Phys Rev E 64(2):021913

    Article  CAS  Google Scholar 

  • Coombes S, Bressloff PC (2005) Bursting: the genesis of rhythm in the nervous system. World Scientific, Singapore

    Book  Google Scholar 

  • Ditlevsen S, Samson A (2013) Introduction to stochastic models in biology. In: Bachar M, Batzel J (eds) Stochastic biomathematical models with applications to neuronal modeling, vol 2058. Lecture notes in mathematics series (biosciences subseries). Springer, Berlin

    Google Scholar 

  • Faisal AA, Selen LPJ, Wolpert DM (2008) Noise in the nervous system. Nat Rev Neurosci 9(4):292

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Gu HG, Jia B, Chen GR (2013) Experimental evidence of a chaotic region in a neural pacemaker. Phys Lett A 377(9):718–720

    Article  CAS  Google Scholar 

  • Gu H, Pan B (2015) A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model. Nonlinear Dyn 81(4):2107–2126

    Article  Google Scholar 

  • Gu H, Pan B, Chen G et al (2014) Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn 78(1):391–407

    Article  Google Scholar 

  • Herrmann CS, Klaus A (2004) Autapse turns neuron into oscillator. Int J Bifurc Chaos 14(2):623–633

    Article  Google Scholar 

  • Herz AVM, Gollisch T, Machens CK et al (2006) Modeling single-neuron dynamics and computations: a balance of detail and abstraction. Science 314(5796):80–85

    Article  CAS  PubMed  Google Scholar 

  • Higham DJ (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 43(3):525–546

    Article  Google Scholar 

  • Hsü ID, Kazarinoff ND (1976) An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model. J Math Anal Appl 55(1):61–89

    Article  Google Scholar 

  • Hsü ID, Kazarinoff ND (1977) Existence and stability of periodic solutions of a third-order non-linear autonomous system simulating immune response in animals. Proc R Soc Edinburgh Sect A 77(1–2):163–175

    Article  Google Scholar 

  • Izhikevich EM (2007) Dynamical systems in neuroscience. MIT Press, Cambridge

    Google Scholar 

  • Izhikevich EM (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 15(5):1063–1070

    Article  PubMed  Google Scholar 

  • Izhikevich EM, Desai NS, Walcott EC et al (2003) Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci 26(3):161–167

    Article  CAS  PubMed  Google Scholar 

  • Kuznetsov YA (1998) Elements of applied bifurcation theory, 2nd edn. Springer, New York

    Google Scholar 

  • Larter R, Speelman B, Worth RM (1999) A coupled ordinary differential lattice model for the simulation of epileptic seizures. Chaos 9(3):795–804

    Article  PubMed  Google Scholar 

  • Lee SG, Kim S (1999) Parameter dependence of stochastic resonance in the stochastic Hodgkin-Huxley neuron. Phys Rev E 60(1):826

    Article  CAS  Google Scholar 

  • Li J, Tang J, Ma J et al (2016) Dynamic transition of neuronal firing induced by abnormal astrocytic glutamate oscillation. Sci Rep 6:32343

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Li Q, Zeng H, Li J (2015) Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn 79(4):2295–2308

    Article  Google Scholar 

  • Lindner B, Garcia-Ojalvo J, Neiman A et al (2004) Effects of noise in excitable systems. Phys Rep 392(6):321–424

    Article  Google Scholar 

  • Longtin A (2010) Stochastic dynamical systems. Scholarpedia 5(4):1619

    Article  Google Scholar 

  • Lv M, Ma J (2016) Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205:375–381

    Article  Google Scholar 

  • Lv M, Wang C, Ren G et al (2016) Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn 85(3):1479–1490

    Article  Google Scholar 

  • Ma J, Tang J (2015) A review for dynamics of collective behaviours of network of neurons. Sci China Technol Sci 58(12):2038–2045

    Article  CAS  Google Scholar 

  • Ma J, Wu F, Hayat T et al (2017a) Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media. Phys A 486:508–516

    Article  Google Scholar 

  • Ma J, Wang Y, Wang C et al (2017b) Mode selection in electrical activities of myocardial cell exposed to electromagnetic radiation. Chaos Solitons Fractals 99:219–225

    Article  Google Scholar 

  • Muthuswamy B (2010) Implementing memristor based chaotic circuits. Int J Bifurc Chaos 20(5):1335–1350

    Article  Google Scholar 

  • Perc M (2006) Thoughts out of noise. Eur J Phys 27(2):451

    Article  Google Scholar 

  • Perc M, Marhl M (2005) Amplification of information transfer in excitable systems that reside in a steady state near a bifurcation point to complex oscillatory behavior. Phys Rev E 71(2):026229

    Article  CAS  Google Scholar 

  • Qin HX, Ma J, Jin WY et al (2014) Dynamics of electrical activities in neuron and neurons of network induced by autapses. Sci China Technol Sci 57(5):936–946

    Article  Google Scholar 

  • Shilnikov A, Kolomiets M (2008) Methods of the qualitative theory for the Hindmarsh-Rose model: a case study-a tutorial. Int J Bifurc Chaos 18(8):2141–2168

    Article  Google Scholar 

  • Song XL, Wang CN, Ma J et al (2015) Transition of electric activity of neurons induced by chemical and electric autapses. Sci China Technol Sci 58(6):1007–1014

    Article  CAS  Google Scholar 

  • Storace M, Linaro D, de Lange E (2008) The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise linear approximations. Chaos 18(3):033128

    Article  PubMed  Google Scholar 

  • Tang J, Liu TB, Ma J et al (2016) Effect of calcium channel noise in astrocytes on neuronal transmission. Commun Nonlinear Sci Numer Simul 32:262–272

    Article  Google Scholar 

  • Tang J, Luo JM, Ma J (2013) Information transmission in a neuron-astrocyte coupled model. PLoS ONE 8(11):e80324

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Tsaneva-Atanasova K, Osinga HM, Rie T et al (2010) Full system bifurcation analysis of endocrine bursting models. J Theor Biol 264(4):1133–1146

    Article  PubMed  PubMed Central  Google Scholar 

  • Upadhyay RK, Mondal A, Teka WW (2017) Mixed mode oscillations and synchronous activity in noise induced modified Morris-Lecar neural system. Int J Bifurc Chaos 27(5):1730019

    Article  Google Scholar 

  • Wang C, Ma J (2018) A review and guidance for pattern selection in spatiotemporal system. Int J Mod Phys B 32(6):1830003

    Article  Google Scholar 

  • Wang Y, Ma J, Xu Y et al (2017) The electrical activity of neurons subject to electromagnetic induction and Gaussian white noise. Int J Bifurc Chaos 27(2):1750030

    Article  Google Scholar 

  • Wig GS, Schlaggar BL, Petersen SE (2011) Concepts and principles in the analysis of brain networks. Ann N Y Acad Sci 1224(1):126–146

    Article  PubMed  Google Scholar 

  • Wu F, Wang C, Xu Y et al (2016a) Model of electrical activity in cardiac tissue under electromagnetic induction. Sci Rep 6(1):28

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Wu H, Bao B, Liu Z et al (2016b) Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn 83(1–2):893–903

    Article  Google Scholar 

  • Xiao-Bo W, Juan M, Ming-Hao Y (2008) Two different bifurcation scenarios in neural firing rhythms discovered in biological experiments by adjusting two parameters. Chin Phys Lett 25(8):2799

    Article  Google Scholar 

  • Xu Y, Ying H, Jia Y et al (2017) Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci Rep 7:43452

    Article  PubMed  PubMed Central  Google Scholar 

  • Zhan F, Liu S (2017) Response of electrical activity in an improved neuron model under electromagnetic radiation and noise. Front Comput Neurosci 11(107):1–12

    Google Scholar 

  • Zhou J, Wu Q, Xiang L (2012) Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization. Nonlinear Dyn 69(3):1393–1403

    Article  Google Scholar 

Download references

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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Correspondence to Ranjit Kumar Upadhyay.

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Appendices

Appendix A

The equilibrium points are derived from the following system of equations

$$\begin{aligned}&({a_1}s - 3\beta ({k_1}/k_2^2)){u^ * }^3 - (s + 1){u^ * }^2 - ({b_1}s({a_2}/k) + {k_1}\alpha ){u^ * } \nonumber \\&\quad - ({b_1}{b_2}/k) = 0, \end{aligned}$$
(4a)
$$\begin{aligned}&{v^{*}} = {u^{ * 2}}, \end{aligned}$$
(4b)
$$\begin{aligned}&{z^ * } = (1/k)(s{a_2}{u^ * } + {b_2}), \end{aligned}$$
(4c)
$$\begin{aligned}&{w^ * } = (1/{k_2}){u^ * }. \end{aligned}$$
(4d)

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

$$\begin{aligned} {J_{E^*}} = \left( {\begin{array}{llll} { - s( - 3{a_1}{u^{ * 2}} + 2{u^ * }) - {k_1}(\alpha + 3\beta {w^{ * 2}})}&{ - 1}&{ - {b_1}}&{ - 6{k_1}\beta {u^ * }{w^ *}} \\ {2{u^ * }}&{ - 1}&0&0\\ {\varepsilon s{a_2}}&0&{ - k\varepsilon }&0\\ 1&0&0& { - {k_2}} \end{array}} \right) . \end{aligned}$$

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

$$\begin{array}{l} {\mu _2}\alpha '({b_{{2_{(crt)}}}}) = \frac{1}{c}[(f_{11}^1 + f_{22}^1)f_{12}^1 - (f_{11}^2 + f_{22}^2)f_{12}^2 - f_{11}^1f_{11}^2 + f_{22}^1f_{22}^2] - [f_{111}^1 + f_{122}^1 + f_{112}^2 + f_{222}^2]\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{2}{{{\mu ^2} + {\psi ^2} + 4{c^2}}}[c\{ (f_{11}^3 - f_{22}^3)(f_{23}^1 + f_{13}^2) + (f_{24}^1 + f_{14}^2)(f_{11}^4 - f_{22}^4)\} + \mu f_{12}^3(f_{23}^1 + f_{13}^2) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \psi f_{12}^4(f_{24}^1 + f_{14}^2)] + \frac{2}{\mu }(f_{11}^3 + f_{22}^3)(f_{13}^1 + f_{23}^2) + \frac{2}{\psi }(f_{11}^4 + f_{22}^4)(f_{14}^1 + f_{24}^2) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{1}{{{\mu ^2} + {\psi ^2} + 4{c^2}}}[(f_{13}^1 - f_{23}^2)\{ \mu (f_{11}^3 - f_{23}^3) - 4cf_{12}^3\} + (f_{14}^1 - f_{24}^2)\{ \psi (f_{11}^4 - f_{22}^4) - 4cf_{12}^4\}], \end{array}$$

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

$$U = \left( {\begin{array}{llll} 0&\quad \beta &\quad 0&\quad 0\\ { - \beta }&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad \mu &\quad 0\\ 0&\quad 0&\quad 0&\quad \psi \end{array}} \right), $$

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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