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Symmetry breaking in solitary solutions to the Hodgkin–Huxley model

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Abstract

This paper presents necessary and sufficient conditions for the existence of bright/dark solitary solutions in the Hodgkin–Huxley model. The second-order analytic solitary solutions are derived using the generalized differential operator technique. It is shown that the heteroclinic bifurcation in the Hodgkin–Huxley model yields a symmetry breaking effect. Trajectories of solitary solutions before the bifurcation lie on manifolds of one of the saddle points and the separatrix between periodic and non-periodic solutions. A new separatrix emerges after the heteroclinic bifurcation—but solitary solutions do not lie on this trajectory. This symmetry breaking effect is demonstrated using analytic and computational experiments.

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Appendices

Apppendix A: Details of the application of the inverse balancing technique to the Hodgkin–Huxley equation

Setting the solitary solution order to \(m = 2\) transforms (26) into:

$$\begin{aligned}&\eta \sigma b \frac{{\widehat{x}}}{{\widehat{c}}} X \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \left( X \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \left( 2 \frac{{\widehat{x}}}{{\widehat{c}}}- y_2-y_1 \right) \right. \nonumber \\&\left. \qquad -\,Y \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \left( 2 \frac{{\widehat{x}}}{{\widehat{c}}}-x_2-x_1 \right) \right) \nonumber \\&\qquad +\,2\eta ^2 \sigma \frac{{\widehat{x}}^2}{{\widehat{c}}^2} \Bigg ( \left( X \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \right) ^{2}- X \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) Y \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \nonumber \\&\qquad -\,X \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \left( 2 \frac{{\widehat{x}}}{{\widehat{c}}}-y_2-y_1 \right) \nonumber \\&\qquad \times \, \left( 2 \frac{{\widehat{x}}}{{\widehat{c}}}-x_2-x_1 \right) +Y \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \left( 2 \frac{{\widehat{x}}}{{\widehat{c}}}-x_2-x_1 \right) ^{2} \Bigg ) \nonumber \\&\quad = \, {\widehat{a}}_{{3}}{\sigma }^{3} \left( Y \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \right) ^{3}+{\widehat{a}}_{{2}}{ \sigma }^{2} \left( Y \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \right) ^{2}X \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \nonumber \\&\qquad + \, {\widehat{a}}_{{1}}\sigma Y \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \left( X \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \right) ^{2}+{\widehat{a}}_{{0}} \left( X \left( \frac{{\widehat{x}}}{{\widehat{c}}}\right) \right) ^{3}.\nonumber \\ \end{aligned}$$
(51)

Taking \({\widehat{x}} = {\widehat{c}} x_1\), \({\widehat{x}} = {\widehat{c}} x_2\), \({\widehat{x}} = {\widehat{c}} y_1\), \({\widehat{x}} = {\widehat{c}} y_2\), \({\widehat{x}} = {\widehat{c}} (x_1+x_2)/2\),\({\widehat{x}} = {\widehat{c}} (y_1+y_2)/2\) and \({\widehat{x}} = 0\) results in seven linear equations. The solutions to these equations with respect to \(b, {\widehat{a}}_0, \dots , {\widehat{a}}_3\) are given as follows:

$$\begin{aligned} {\widehat{a}}_3= & {} 2 \left( \frac{\eta x_1 \left( x_1 - x_2\right) }{\sigma Y\left( x_1\right) }\right) ^2; \end{aligned}$$
(52)
$$\begin{aligned} {\widehat{a}}_0= & {} -\frac{2y_1y_2\sigma \eta ^2 \left( \varTheta _1 + \varTheta _2\right) }{\varOmega _1}, \,\, \nonumber \\ b= & {} -\eta + \frac{2\left( y_2 X\left( y_1\right) \varTheta _2 - y_1 X\left( y_2\right) \varTheta _1\right) }{\left( y_2 - y_1\right) \varOmega _1} , \end{aligned}$$
(53)

where

$$\begin{aligned} \varTheta _1:= & {} y_1 X\left( y_2\right) \left( \left( y_2 - y_1\right) \left( x_1 + x_2 - 2y_2\right) - X\left( y_1\right) \right) ;\nonumber \\ \varTheta _2:= & {} y_2 X\left( y_1\right) \left( \left( y_1 - y_2\right) \left( x_1 + x_2 - 2y_1\right) - X\left( y_2\right) \right) ;\nonumber \\ \varOmega _1:= & {} X\left( y_1\right) X\left( y_2\right) \left( y_2 X\left( y_1\right) + y_1 X\left( y_2\right) \right) . \end{aligned}$$
(54)
$$\begin{aligned} {\widehat{a}}_1= & {} \frac{1}{\varOmega _2} \left( \varLambda _{10} {\widehat{a}}_0 + \varLambda _{13} {\widehat{a}}_3 + \varLambda _3 \left( 2\tau _x X\left( \tau _x\right) Y\left( \tau _y\right) \right. \right. \nonumber \\&\left. \left. \quad - \, 2\tau _y Y^2\left( \tau _x\right) \right) \left( \eta ^2 + \eta b\right) + \sigma \eta ^2 X\left( \tau _x\right) \right. \nonumber \\&\quad \left. \times \Big (Y\left( \tau _y\right) \varPhi _1 - Y\left( \tau _x\right) \Big (\varPhi _2 + 2 \tau _x^2 X\left( \tau _y\right) Y^2\left( \tau _y\right) \right. \nonumber \\&\quad \left. + \, 2 \tau _y^2 Y\left( \tau _x\right) X^2\left( \tau _y\right) \Big ) \Big ) \right) ;\nonumber \\ {\widehat{a}}_2= & {} \frac{1}{\sigma \varOmega _2} \left( -\varLambda _{20} {\widehat{a}}_0 - \varLambda _{23} {\widehat{a}}_3 - \varLambda _3 X\left( \tau _x\right) \left( 2 \tau _x X\left( \tau _y\right) \right. \right. \nonumber \\&\left. \left. \quad -\, 2 \tau _y Y\left( \tau _x\right) \right) \left( \eta ^2 + \eta b\right) + \sigma \eta ^2 X\left( \tau _x\right) \right. \nonumber \\&\quad \left. \times \Big (Y\left( \tau _x\right) X\left( \tau _y\right) \varPhi _1 + X^2\left( \tau _x\right) \right. \nonumber \\&\left. \qquad \Big (\varPhi _2 + 2\tau _y^2 X^2 \left( \tau _y\right) Y\left( \tau _x\right) \right. \nonumber \\&\quad \left. - \, 2 \tau _x^2 X^2\left( \tau _y\right) Y\left( \tau _y\right) \Big )\Big ) \right) , \end{aligned}$$
(55)

where

$$\begin{aligned} \tau _x&:= \frac{x_1 + x_2}{2}, \quad \tau _y := \frac{y_1 + y_2}{2}, \quad \\ \varPhi _1&:= 2\tau _x^2 X\left( \tau _y\right) X\left( \tau _x\right) Y\left( \tau _y\right) ;\\ \varPhi _2&:= 2\tau _y^2 Y\left( \tau _x\right) Y\left( \tau _y\right) \left( 4\left( \tau _x-\tau _y\right) ^2 - X\left( \tau _y\right) \right) ;\\ \varLambda _{10}&:= X\left( \tau _x\right) Y^2\left( \tau _x\right) X^3 \left( \tau _y\right) \\ {}&\quad - X\left( \tau _y\right) Y^2\left( \tau _y\right) X^3\left( \tau _x\right) ;\\ \varLambda _{13}&:= \sigma ^3\left( X\left( \tau _x\right) Y^2\left( \tau _x\right) Y^3\left( \tau _y\right) \right. \\&\left. \quad - X\left( \tau _y\right) Y^2\left( \tau _y\right) Y^3\left( \tau _x\right) \right) ;\\ \varLambda _{20}&:= Y\left( \tau _y\right) X^2\left( \tau _y\right) X^3\left( \tau _x\right) \\ {}&\quad -Y\left( \tau _x\right) X^2\left( \tau _y\right) X^3\left( \tau _y\right) ;\\ \varLambda _{23}&:= \sigma ^3 \left( Y\left( \tau _y\right) X^2\left( \tau _y\right) Y^3\left( \tau _x\right) \right. \\&\left. \quad - Y\left( \tau _y\right) X^2\left( \tau _x\right) Y^3\left( \tau _y\right) \right) ;\\ \varLambda _3&:= 2\sigma \eta X\left( \tau _y\right) X\left( \tau _x\right) Y\left( \tau _y\right) \left( \tau _x - \tau _y\right) ;\\ \varOmega _2&:= \sigma X\left( \tau _x\right) X\left( \tau _y\right) Y\left( \tau _x\right) Y\left( \tau _y\right) \\&\quad \left( Y\left( \tau _y\right) X\left( \tau _x\right) - Y\left( \tau _x\right) X\left( \tau _y\right) \right) . \end{aligned}$$

Appendix B: Construction of solitary solutions to (41)

An independent variable transformation (9) is applied to (41):

$$\begin{aligned} \eta {\widehat{x}} {\widehat{z}}'_{{\widehat{z}}} = A({\widehat{z}}-z_1) \sqrt{({\widehat{z}}-z_2)({\widehat{z}}-z_3)}. \end{aligned}$$
(56)

Initial conditions (37) are transformed to:

$$\begin{aligned} {\widehat{z}}\left( {\widehat{c}}\right) = u. \end{aligned}$$
(57)

The generalized differential operator with respect to (56) reads:

$$\begin{aligned} \varvec{\mathrm {D}} := \varvec{\mathrm {D}}_{{\widehat{c}}} + \frac{1}{\eta {\widehat{c}}} A \left( u - z_1\right) \sqrt{\left( u - z_2\right) \left( u-z_3\right) } \varvec{\mathrm {D}}_u.\nonumber \\ \end{aligned}$$
(58)

Computing Hankel determinants (16) for the sequence \({\widehat{p}}_j := \frac{1}{j!} \varvec{\mathrm {D}}^j u; \, j = 0,1,\dots \), yields the following relation:

$$\begin{aligned} d_4 = \frac{1}{\eta ^{12} {\widehat{c}}^{12}} \left( \eta ^2 - A^2 \left( z_1-z_2\right) \left( z_1-z_3\right) \right) f\left( \eta , u\right) ,\nonumber \\ \end{aligned}$$
(59)

where \(f\left( \eta , u\right) \) is a polynomial in \(\eta , u\). The above equation yields that \(d_4 = 0\) only if:

$$\begin{aligned} \eta = \pm A \sqrt{\left( z_1-z_2\right) \left( z_1-z_3\right) }. \end{aligned}$$
(60)

Equation (56) can admit closed-form solutions only if conditions (19) hold true. To verify (19), the characteristic roots \({\widehat{\rho }}_k, k = 1,2,3\), are computed from the characteristic polynomial:

$$\begin{aligned} \begin{vmatrix} {\widehat{p}}_0&{\widehat{p}}_1&{\widehat{p}}_2&{\widehat{p}}_3 \\ {\widehat{p}}_1&{\widehat{p}}_2&{\widehat{p}}_3&{\widehat{p}}_4 \\ {\widehat{p}}_2&{\widehat{p}}_3&{\widehat{p}}_4&{\widehat{p}}_5 \\ 1&{\widehat{\rho }}&{\widehat{\rho }}^2&{\widehat{\rho }}^3 \\ \end{vmatrix} = 0. \end{aligned}$$
(61)

Solution to (61) reads:

$$\begin{aligned} {\widehat{\rho }}_1&= 0; \end{aligned}$$
(62)
$$\begin{aligned} {\widehat{\rho }}_2&= \frac{1}{2{\widehat{c}} \left( z_1-z_3\right) \left( z_1-z_2\right) } \nonumber \\&\quad \Big (\left( \pm \sqrt{ \left( u-z_2\right) \left( u-z_3\right) } + u - z_1\right) \nonumber \\&\quad \times \sqrt{ \left( z_1 - z_2\right) \left( z_1 - z_3\right) } - \left( z_1 - z_2\right) \left( z_1 - z_3\right) \Big ); \end{aligned}$$
(63)
$$\begin{aligned} {\widehat{\rho }}_3&= \frac{1}{2{\widehat{c}} \left( z_1-z_3\right) \left( z_1-z_2\right) } \nonumber \\&\quad \Big (\left( \pm \sqrt{ \left( u-z_2\right) \left( u-z_3\right) } - u + z_1\right) \nonumber \\&\quad \times \sqrt{ \left( z_1 - z_2\right) \left( z_1 - z_3\right) } - \left( z_1 - z_2\right) \left( z_1 - z_3\right) \Big ). \end{aligned}$$
(64)

Formula (18) together with (62)–(64) yields:

$$\begin{aligned} {\widehat{p}}_j = \lambda _1 0^j + \lambda _2 {\widehat{\rho }}_1^j + \lambda _3 {\widehat{\rho }}_2^j. \end{aligned}$$
(65)

Note that \(0^0 := 1\). Definition of the generalized differential operator \(\varvec{\mathrm {D}}\) and sequence \({\widehat{p}}_j\) for \(j = 0,1,2\) results in:

$$\begin{aligned} \lambda _1&= z_1; \end{aligned}$$
(66)
$$\begin{aligned} \lambda _2&= \frac{{\widehat{p}}_1 {\widehat{\rho }}_2 - {\widehat{p}}_2}{{\widehat{\rho }}_1 \left( {\widehat{\rho }}_2 - {\widehat{\rho }}_1\right) }; \end{aligned}$$
(67)
$$\begin{aligned} \lambda _3&= \frac{{\widehat{p}}_1 {\widehat{\rho }}_1 - {\widehat{p}}_2}{{\widehat{\rho }}_2 \left( {\widehat{\rho }}_1 - {\widehat{\rho }}_2\right) }. \end{aligned}$$
(68)

Using (62)–(64) and (66)–(68), it is verified that conditions (19) hold true; thus, the solution to (56) reads:

$$\begin{aligned} {\widehat{z}} = z_1 + \frac{\lambda _2}{1-{\widehat{\rho }}_2\left( {\widehat{x}}- {\widehat{c}}\right) } + \frac{\lambda _3}{1-{\widehat{\rho }}_3\left( {\widehat{x}}- {\widehat{c}}\right) }. \end{aligned}$$
(69)

Using the inverse of transformation (9) yields the solution to (41):

$$\begin{aligned} z = \frac{z_1 \left( \exp \left( \eta \left( x-c\right) \right) - \alpha _1\right) \left( \exp \left( \eta \left( x-c\right) \right) - \alpha _2\right) }{\left( \exp \left( \eta \left( x-c\right) \right) - 1 - \frac{1}{\rho _2}\right) \left( \exp \left( \eta \left( x-c\right) \right) - 1 - \frac{1}{\rho _3}\right) },\nonumber \\ \end{aligned}$$
(70)

where

$$\begin{aligned} \rho _k = \rho _k(u) = {\widehat{c}}{\widehat{\rho }}_k; \quad k = 2,3; \end{aligned}$$
(71)

and

$$\begin{aligned} \alpha _2= & {} \frac{1}{2z_1} \left( -K+\sqrt{K^2-4z_1L}\right) ; \nonumber \\ \alpha _3= & {} -\frac{1}{2z_1} \left( K+\sqrt{K^2-4z_1L}\right) . \end{aligned}$$
(72)

The functions \(K(u), L(u)\) have the following expressions:

$$\begin{aligned} K&= -\left( z_1\left( 2 + \frac{1}{\rho _2} + \frac{1}{\rho _3}\right) + \frac{\mu _2}{\rho _2} + \frac{\mu _3}{\rho _3} \right) ; \end{aligned}$$
(73)
$$\begin{aligned} L&= z_1 \left( 1 + \frac{1}{\rho _2}\right) \left( 1 + \frac{1}{\rho _3}\right) \nonumber \\&\quad + \frac{\mu _2}{\rho _2} \left( 1 + \frac{1}{\rho _3}\right) + \frac{\mu _3}{\rho _3} \left( 1 + \frac{1}{\rho _2}\right) . \end{aligned}$$
(74)

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Telksnys, T., Navickas, Z., Timofejeva, I. et al. Symmetry breaking in solitary solutions to the Hodgkin–Huxley model. Nonlinear Dyn 97, 571–582 (2019). https://doi.org/10.1007/s11071-019-04998-4

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