Continuous signaling pathways instability in an electromechanical coupled model for biomembranes and nerves

Abstract

Information processing is a basic part of life and, presumably, the nervous system’s primary purpose. Neurons perform a variety of tasks that collect useful information from the organism’s periphery sensor receptor arrays and transfer it into action, imagery, and memory. Brain communication, also known as neural signaling, is now assumed to be electromechanical, based on a large body of observational and experimental evidence acquired over the last two centuries. In this paper, modulational instability is investigated as a mechanism of wave trains and soliton formation in neurons using a nonlinearly coupled complex Ginzburg–Landau equation derived from the Morris–Lecar neural model for nerve electrical activity and the Heimburg–Jackson model for longitudinal density pulses. Using standard linear stability analysis, the growth rate of modulation instability is studied analytically and numerically as a function of perturbation frequency and system parameters. In particular, it is shown that the gain depends strongly on the coupling parameters. We also investigated the effect of coupling parameters on the gain of modulation instability, thus confirming the continuous signaling instability. This is highly significant from a theoretical point of view and could be a plus to explain the process of generation of action potential as the consequences of instability between coupled electrical and mechanical weakly continuous wave in nerve.

Graphical abstract

Appendices

Appendix A

The following are expressions of parameters used in Eq. (12):

$$\begin{aligned} {\eta _0}= & {} {a_2}^2{b_7} - {a_2}{b_2}{a_6} + {a_2}{b_2}{\sigma _1},\nonumber \\ {\eta _1}= & {} - {a_2}{b_2}{a_1} + {a_2}^2{b_1} + 2{a_2}{a_4}{b_7} - {a_2}{b_4}{a_6} + \frac{{{a_2}{b_2}}}{{{v_0}}} + {a_2}{b_4}{\sigma _1} - {a_4}{b_2}{a_6} + {a_4}{b_2}{\sigma _1}, \nonumber \\ {\eta _2}= & {} - {a_2}{b_4}{a_1} - {a_4}{b_2}{a_1} + {a_2}^2{b_3} - {a_2}{b_2}{a_3} + 2{a_2}{a_4}{b_1} - {a_2}{b_6}{a_6}\nonumber \\&\quad + \frac{{{a_2}{b_4}}}{{{v_0}}} + {a_2}{b_6}{\sigma _1} + {a_4}^2{b_7} - {a_4}{b_4}{a_6} + \frac{{{a_4}{b_2}}}{{{v_0}}} + {a_4}{b_4}{\sigma _1}, \nonumber \\ {\eta _3}= & {} - {a_2}{b_6}{a_1} - {a_4}{b_4}{a_1} + {a_2}^2{b_5} - {a_2}{b_4}{a_3} + 2{a_2}{a_4}{b_3} - {a_2}{b_2}{a_5}\nonumber \\&\quad + \frac{{{a_2}{b_6}}}{{{v_0}}} - {a_4}{b_2}{a_3} + {a_4}^2{b_1} - {a_4}{b_6}{a_6} + \frac{{{a_4}{b_4}}}{{{v_0}}} + {a_4}{b_6}{\sigma _1}, \nonumber \\ {\kappa _0}= & {} - \frac{{{a_2}{b_2}}}{{\rho _0^A}},\,\,{\kappa _1} = - \frac{{{a_2}{b_2}}}{{\rho _0^A{v_0}}} - \frac{{{a_2}{b_4}}}{{\rho _0^A}} - \frac{{{a_4}{b_2}}}{{\rho _0^A}},\,\,{\kappa _2} = - \frac{{{a_2}{b_4}}}{{\rho _0^A{v_0}}} - \frac{{{a_2}{b_6}}}{{\rho _0^A}} - \frac{{{a_4}{b_2}}}{{\rho _0^A{v_0}}} - \frac{{{a_4}{b_4}}}{{\rho _0^A}}, \nonumber \\ {\beta _0}= & {} {a_2}{b_2}{C_{{m_0}}} + {a_1}{a_2} - \frac{{{a_2}}}{{\rho _0^A{v_0}}} - {a_4}{a_6} + {a_4}{\sigma _1},\,\,\,\nonumber \\ {\beta _1}= & {} {a_2}{b_4}{C_{{m_0}}} + {a_4}{b_2}{C_{{m_0}}} + 2{a_2}{a_3}, \nonumber \\ {\beta _2}= & {} {a_2}{b_6}{C_{{m_0}}} + {a_4}{b_4}{C_{{m_0}}} + 3{a_2}{a_5} + {a_4}{a_3},\,\,\,{\beta _4} = \frac{{{a_2}}}{{\rho _0^A{v_0}}} - \frac{{{a_4}}}{{\rho _0^A}} \nonumber \\ {\gamma _0}= & {} - {a_2}{b_2}D,\,\,\,{\gamma _1} = - D({a_2}{b_4} + {a_4}{b_2}),\,\,{\gamma _2} = - D({a_2}{b_6} + {a_4}{b_4}),\, \nonumber \\ {\delta _1}= & {} {C_{{m_0}}}{a_4},\,{\delta _2} = {C_{{m_0}}}{a_2},\,\,\,{\delta _3} = - D{a_4},\,\,{\delta _4} = D{a_2},\,\,{\delta _5} = \frac{{{a_2}}}{{\rho _0^A{v_0}}} + \frac{{{a_4}}}{{\rho _0^A}},\,\,{\delta _6} = \frac{{{a_4}}}{{\rho _0^A{v_0}}},\,\,{\delta _7} = \frac{{{a_2}}}{{\rho _0^A}} \nonumber \\ {a_1}= & {} \frac{{{g_{{c_a}}}{V_1}^3 {-} 3{V_1}^2{V_{{c_a}}}{g_{{c_a}}} {+} 3{V_1}{V_2}^2{g_{{c_a}}} {-} 6{g_L}{V_2}^3 {-} 3{g_{{c_a}}}{V_2}^3 {+} 3{V_2}^2{V_{{c_a}}}{g_{{c_a}}}}}{{6{V_2}^3}}, \nonumber \\ {a_2}= & {} {g_K}{V_K},\,\,{a_3} = \frac{{{g_{{c_a}}}\left( {{V_1}^2 + {V_1}{V_{{c_a}}} - 2{V_2}^2} \right) }}{{4{V_2}^3}},\,\,\nonumber \\ {a_4}= & {} - {g_K},\,\,{a_5} = - \frac{{{g_{{c_a}}}\left( {{V_{{c_a}}} + 3{V_1}} \right) }}{{6{V_2}^3}}, \nonumber \\ {a_6}= & {} \frac{{{g_{{c_a}}}{V_1}^3{V_{{c_a}}} - 3{g_{{c_a}}}{V_1}{V_{{c_a}}}{V_2}^2 + 6{g_L}{V_L}{V_2}^3 + 3{g_{{c_a}}}{V_{{c_a}}}{V_2}^3}}{{6{V_2}^3}}, \nonumber \\ {b_1}= & {} - \frac{{5{V_3}^4 + 15{V_3}^2{V_4}^2 + 6{V_3}{V_4}^3 - 24{V_4}^4}}{{144{V_4}^5}},\,\,\nonumber \\ {b_2}= & {} - \frac{{{V_3}^2}}{{24{V_4}^2}} - \frac{1}{3}, \nonumber \\ {b_3}= & {} \frac{{10{V_3}^3 + 15{V_3}{V_4}^2 + 3{V_4}^3}}{{144{V_4}^5}},\,\,{b_4} = \frac{{{V_3}}}{{12{V_4}^2}},\,\,\nonumber \\ {b_5}= & {} - \frac{1}{{144{V_4}^3}} - \frac{{5{V_3}^2}}{{72{V_4}^5}}, {b_6} = - \frac{1}{{24{V_4}^2}},\,\,\,\nonumber \\ {b_7}= & {} \frac{{{V_3}^5 + 5{V_3}^3{V_4}^2 + 3{V_3}^2{V_4}^3 - 24{V_3}{V_4}^4 + 24{V_4}^5}}{{144{V_4}^5}}. \end{aligned}$$

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Appendix B

$$\begin{aligned} {P_1}= & {} \frac{{v_g^2{\delta _2} - {\gamma _0}}}{{{\delta _2}\omega }},\,\,\nonumber \\ {P_2}= & {} \frac{{{c_g}^2({k^2}{H_2} + 1) + 4k\omega {H_2}{c_g} - 6{k^2}{H_1} + {\omega ^2}{H_2} - {c_0}^2}}{{\left( {{k^2}{H_2} + 1} \right) \omega }},\,\,\nonumber \\ {Q_{{1_r}}}= & {} \frac{{ - {k^4}{\gamma _1}^2 + {k^2}\left( {9{\delta _3}{\omega ^2}{\beta _1} + 9{\delta _1}{\omega ^2}{\gamma _1} - 9{\gamma _2}{\eta _1} + 11{\eta _2}{\gamma _1}} \right) + 9{\eta _3}{\eta _1} - 12{\delta _1}^2{\omega ^4} - {\omega ^2}{\beta _1}^2 - 10{\eta _2}^2 + 21{\delta _1}{\omega ^2}{\eta _2}}}{{6{\eta _1}{\delta _2}\omega }}, \end{aligned}$$

$$\begin{aligned} {Q_{{1_i}}}= & {} - \frac{{{k^4}{\delta _3}{\gamma _1} + 4{\omega ^2}{k^2}{\delta _1}{\delta _3} - {k^2}{\delta _3}{\eta _2} - {\omega ^2}{\beta _1}{\delta _1} - {\beta _1}{\eta _2} + {\beta _2}{\eta _1}}}{{2{\eta _1}{\delta _2}}},\,\,\nonumber \\ {Q_1}= & {} {Q_{{1_r}}} + i{Q_{{1_i}}},\,\,{R_1} = \frac{{{k^2}{\delta _4} - {\beta _0}}}{{2{\delta _2}\omega }},\,\,{K_{{1_r}}} = \frac{1}{2}\frac{{{\kappa _0}}}{{{\delta _2}\omega }},\,\, \nonumber \\ {Q_2}= & {} \frac{{{\alpha ^2}\left( {{k^2}{H_2} + 1} \right) - 6\beta {k^2}\left( {{H_1} - {H_2}{c_0}^2} \right) }}{{12\left( {{H_1} - {H_2}{c_0}^2} \right) \left( {{k^2}{H_2} + 1} \right) \omega }},\,\,\nonumber \\ {R_2}= & {} \frac{{\vartheta {k^2}}}{{2({k^2}{H_2} + 1)}},\,\,\,{K_2} = \frac{{{\sigma _2}}}{{2\left( {{k^2}{H_2} + 1} \right) \omega }}{K_{{1_i}}}\nonumber \\= & {} - \frac{{{\delta _7}}}{{2{\delta _2}}},\,\,{K_1} = {K_{{1_i}}} + i{K_{{1_i}}}, \nonumber \\ {M_1}= & {} - \frac{{{\omega ^2}{\beta _4}^2 - {\omega ^2}{\beta _4}{\delta _5} + {\omega ^2}{\delta _5}^2 + {\kappa _1}^2}}{{3{\eta _1}{\delta _2}\omega }}, \nonumber \\ {N_{{1_r}}}= & {} \frac{{6{k^2}{\omega ^2}{\beta _4}{\delta _3} + 3{k^2}{\omega ^2}{\delta _3}{\delta _5} + 2{k^2}{\gamma _1}{\kappa _1} - 2{\omega ^2}{\beta _1}{\beta _4} + {\omega ^2}{\beta _1}{\delta _5} - 3{\omega ^2}{\delta _1}{\kappa _1} + 3{\eta _1}{\kappa _2} - 5{\eta _2}{\kappa _1}}}{{3{\eta _1}{\delta _2}\omega }}, \nonumber \\ {N_{{1_i}}}= & {} \frac{{4{k^2}{\beta _4}{\gamma _1} + 3{k^2}{\delta _3}{\kappa _1} - 2{k^2}{\delta _5}{\gamma _1} - 6{\omega ^2}{\beta _4}{\delta _1} + 3{\omega ^2}{\delta _1}{\delta _5} + {\beta _1}{\kappa _1} - 4{\beta _4}{\eta _2} + 5{\delta _5}{\eta _2} - 3{\delta _6}{\eta _1}}}{{3{\eta _1}{\delta _2}}}, \nonumber \\ {N_1}= & {} {N_{{1_r}}} + i{N_{{1_i}}},\,{\Gamma _r} = \frac{{6\left( {{\beta _4}{\delta _5} - {\delta _5}^2} \right) \left( {{H_2}{c_0}^2 - {H_1}} \right) {k^2}{\omega ^2} + \left( {6{H_2}{c_0}^2{\kappa _1}^2 + \alpha {H_2}{\eta _1}{\kappa _1} + 6{H_1}{\kappa _1}^2} \right) {k^2} - \alpha {\eta _1}{\kappa _1}}}{{12{\eta _1}\left( { - {H_2}{c_0}^2 + {H_1}} \right) {k^2}{\delta _2}\omega }}, \nonumber \\ {\Gamma _i}= & {} \frac{{6{H_2}\left( {{\beta _4} - 2{\delta _5}} \right) {\kappa _1}{k^2}{c_0}^2 - \left( {\left( {{\beta _4} - 2{\delta _5}} \right) \left( {\alpha {\eta _1}{H_2} + 6{H_1}{\kappa _1}} \right) } \right) {k^2} - \alpha {\eta _1}\left( {{\beta _4} - 2{\delta _5}} \right) }}{{12{\eta _1}\left( { - {H_2}{c_0}^2 + {H_1}} \right) {k^2}{\delta _2}}},\Gamma = {\Gamma _r} + {\Gamma _i},\,\, \nonumber \\ {\chi _r}= & {} \frac{{3\left( {{\delta _5} - {\beta _4}} \right) {\delta _3}{k^2}{\omega ^2} + \left( {{\beta _1}{\beta _4} - 2{\beta _1}{\delta _5} - 3{\delta _1}{\kappa _1}} \right) {\omega ^2} + 2{k^2}{\gamma _1}{\kappa _1} - 5{\eta _2}{\kappa _1}}}{{6{\eta _1}{\delta _2}\omega }}, \nonumber \\ {\chi _{\text {i}}}= & {} \frac{{ - \left( {{\beta _4}{\gamma _1} + 3{\delta _3}{\kappa _1}} \right) {k^2} + 3{\omega ^2}\left( {{\beta _4} - {\delta _5}} \right) {\delta _1} + 2{\beta _1}{\kappa _1} + 4{\beta _4}{\eta _2} - 5{\delta _5}{\eta _2} + 3{\delta _6}{\eta _1}}}{{6{\eta _1}{\delta _2}}},\,\,\chi = {\chi _r} + {\chi _i}. \end{aligned}$$

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Appendix C

$$\begin{aligned} {\Sigma _1}= & {} \frac{{\left( {{K_2}\left( {{N_{{1_i}}} + 3{K_2}{\chi _{\text {i}}}} \right) {N_{{i_1}}}^2 + \left( {3{N_{{1_i}}} + {\chi _{\text {i}}}} \right) {\chi _{\text {i}}}^2{K_2} + 2{K_{{1_i}}}\left( {{N_{{1_i}}} + {\chi _{\text {i}}}} \right) {Q_{{1_i}}}{Q_{{1_r}}} - \left( {{K_{{1_i}}}\left( {{N_{{1_r}}} + {\chi _r}} \right) + {K_{{1_r}}}\left( {{\chi _i} + {N_{{1_i}}}} \right) } \right) Q_{{1_i}}^2} \right) {e^{2{R_1}T}}}}{{{Q_{{1_i}}}\left( {\left( {{\mathrm{T}_r} + {M_1} - {Q_2}} \right) {Q_{{1_i}}}^2 + \left( {N_{{1_i}}^2 + 2{N_{{1_i}}}{\chi _{\text {i}}} + {\chi _{\text {i}}}^2} \right) {Q_{{1_r}}} - \left( {{N_{{1_i}}}{N_{{1_r}}} + {N_{{1_r}}}{\chi _{\text {i}}} + {\chi _{\text {i}}}{\chi _r} + {N_{{1_i}}}{\chi _r}} \right) {Q_{{1_i}}}} \right) }}, \nonumber \\ {\Sigma _2}= & {} \frac{{{e^{4{R_1}T}}{K_{{1_i}}}\left( {3{K_2}{N_{{1_i}}}^2 + 6{K_2}{N_{{1_i}}}{\chi _{\text {i}}} + 3{K_2}{\chi _{\text {i}}}^2 + {K_{{1_i}}}{Q_{{1_i}}}{Q_{{1_r}}} - {K_{{1_r}}}{Q_{{1_i}}}^2} \right) }}{{{Q_{{1_i}}}\left( {\left( {{\mathrm{T}_r} + {M_1} - {Q_2}} \right) {Q_{{1_i}}}^2 + \left( {N_{{1_i}}^2 + 2{N_{{1_i}}}{\chi _{\text {i}}} + {\chi _{\text {i}}}^2} \right) {Q_{{1_r}}} - \left( {{N_{{1_i}}}{N_{{1_r}}} + {N_{{1_r}}}{\chi _{\text {i}}} + {\chi _{\text {i}}}{\chi _r} + {N_{{1_i}}}{\chi _r}} \right) {Q_{{1_i}}}} \right) }}, \nonumber \\ {\Sigma _3}= & {} \frac{{3{e^{6{R_1}T}}{K_2}{K_{{1_i}}}^2\left( {{N_{{1_i}}} + {\chi _{\text {i}}}} \right) }}{{{Q_{{1_i}}}\left( {\left( {{\mathrm{T}_r} + {M_1} - {Q_2}} \right) {Q_{{1_i}}}^2 + \left( {N_{{1_i}}^2 + 2{N_{{1_i}}}{\chi _{\text {i}}} + {\chi _{\text {i}}}^2} \right) {Q_{{1_r}}} - \left( {{N_{{1_i}}}{N_{{1_r}}} + {N_{{1_r}}}{\chi _{\text {i}}} + {\chi _{\text {i}}}{\chi _r} + {N_{{1_i}}}{\chi _r}} \right) {Q_{{1_i}}}} \right) }}, \nonumber \\ {\Sigma _4}= & {} \frac{{{e^{8{R_1}T}}{K_2}{K_{{1_i}}}^3}}{{{Q_{{1_i}}}\left( {\left( {{\mathrm{T}_r} + {M_1} - {Q_2}} \right) {Q_{{1_i}}}^2 + \left( {N_{{1_i}}^2 + 2{N_{{1_i}}}{\chi _{\text {i}}} + {\chi _{\text {i}}}^2} \right) {Q_{{1_r}}} - \left( {{N_{{1_i}}}{N_{{1_r}}} + {N_{{1_r}}}{\chi _{\text {i}}} + {\chi _{\text {i}}}{\chi _r} + {N_{{1_i}}}{\chi _r}} \right) {Q_{{1_i}}}} \right) }}, \nonumber \\ {P_0}= & {} \sqrt{{{\left( {27\Sigma _1^2{\Sigma _4} - 9{\Sigma _1}{\Sigma _2}{\Sigma _3} + 2\Sigma _2^3 - 72{\Sigma _2}{\Sigma _4} + 27\Sigma _3^2} \right) }^2} - 4{{\left( { - 3{\Sigma _1}{\Sigma _3} + \Sigma _2^2 + 12{\Sigma _4}} \right) }^3}}, \nonumber \\ {p_1}= & {} 2^{-1/3}\root 3 \of {{{P_0} + 27\Sigma _1^2{\Sigma _4} - 9{\Sigma _1}{\Sigma _2}{\Sigma _3} + 2\Sigma _2^3 - 72{\Sigma _2}{\Sigma _4} + 27\Sigma _3^2}}, \nonumber \\ {p_2}= & {} \sqrt{\frac{{\Sigma _1^2}}{4} + \frac{{ - 3{\Sigma _1}{\Sigma _3} + \Sigma _2^2 + 12{\Sigma _4}}}{{3{p_1}}} - \frac{{2{\Sigma _2}}}{3} + \frac{{{P_1}}}{3}.} \end{aligned}$$

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Appendix D

$$\begin{aligned} {\lambda _1}= & {} \chi {\Sigma _8}{\psi _{10}}{\psi _{20}} + \frac{1}{2}{\Omega ^2}{P_1} - \Gamma {\Sigma _6}\psi _{20}^2 - \frac{{{K_1}{\Sigma _7}{\psi _{20}}}}{{{\psi _{10}}}} + {Q_1}{\Sigma _5}\psi _{10}^2, \nonumber \\ {\lambda _2}= & {} \Gamma {\Sigma _6}\psi _{20}^2 + {N_1}{\Sigma _8}{\psi _{10}}{\psi _{20}} + {Q_1}{\Sigma _5}\psi _{10}^2, \nonumber \\ {\lambda _3}= & {} 2\Gamma {\Sigma _6}{\psi _{10}}{\psi _{20}} + {M_1}{\Sigma _6}{\psi _{10}}{\psi _{20}} + {N_1}{\Sigma _8}\psi _{10}^2 + {K_1}{\Sigma _7}, \nonumber \\ {\lambda _4}= & {} \chi {\Sigma _8}\psi _{10}^2 + {M_1}{\Sigma _6}{\psi _{10}}{\psi _{20}}, \nonumber \\ {\lambda _5}= & {} {\Sigma _9}{K_2}, \nonumber \\ {\lambda _7}= & {} {Q_2}{\Sigma _6}\psi _{20}^2 + \frac{1}{2}{\Omega ^2}{P_2} - \frac{{{\Sigma _9}{K_2}{\psi _{10}}}}{{{\psi _{20}}}}, \nonumber \\ {\lambda _8}= & {} {Q_2}{\Sigma _6}\psi _{20}^2. \end{aligned}$$

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