Traveling Waves Impulses of FitzHugh Model with Diffusion and Cross-Diffusion

Abstract

The FitzHugh equations have been used as a caricature of the Hodgkin–Huxley equations of neuron firing and to capture, qualitatively, the general properties of an excitable membrane. The spatial propagation of neuron firing due to diffusion of the current potential was described by the FitzHugh–Nagumo model. Assuming that the spatial propagation of neuron firing is caused by not diffusion but cross-diffusion connection between the potential and recovery variables the cross-diffusion version of the FitzHugh model gives rise to the typical fast traveling wave solutions characteristic to the FitzHugh model, and additionally gives rise to the slow traveling wave solutions exhibited in the diffusion FitzHugh–Nagumo equations (Berezovskaya et al., Math Biosci Eng 5:239–260, 2008).

In this paper the FitzHugh model with both diffusion and cross-diffusion terms is studied; it is shown that this new version of spatial FitzHugh model gives rise to fast and slow traveling impulses.

Appendix

1.1.1 Lienard Form of the FitzHugh Model andItsWaveSystem

Through the change of variables

$$ \begin{aligned}[b] &\left(P,Q\right)\to \left(U,V\right):U=Q+{k}_2,\ V={F}_2\left(P,Q\right) \equiv {k}_1P - Q - {k}_2,\\ &\quad\left(P=\left(U+V\right)/{k}_1,\;Q=U-{k}_2,\;{k}_1\ne 0\right)\end{aligned} $$

(1.9)

the local model (1.1) is transformed to the generalized Lienard form:

$$ \begin{aligned}[b]{U}_t&=V,{} e{V}_t=\left(U+V\right)/{k}_1-{\left(U+V\right)}^3/{k_1}^3\\ &\quad+{k}_2-U\equiv f(U) + V\left({g}_1(U)+VG(U)\right)\equiv \varPhi \left(U,V\right),\end{aligned} $$

(1.10)

Where

$$ \begin{array}{l}f(u)=-{u}^3/{k}_1^2 + u\left(1 - {k}_1\right) + {k}_1{k}_2,\\ {}{g}_1(u) = \left(1-e\right)-3{u}^2/{k}_1^2, \\ {}G\left(u,v\right) = -\left(3u+v\right)/{k}_1^2\end{array} $$

(1.11)

Model (1.2) after transformation (1.10) reads

$$ \begin{array}{l}{U}_t=V,\\ {}e{V}_t=\varPhi \left(U,V\right)+{D}_P{V}_{xx}+\left({D}_P+D{}_Qk_1\right){U}_{xx}\end{array} $$

(1.12)

Model (1.3) after transformation (1.10) reads

$$ \begin{array}{l}{U}_t=V,\\ {}e{V}_t=\varPhi \left(U,V\right)+{D}_Q{k}_1{U}_{xx}\end{array} $$

(1.13)

A traveling wave solution of systems (1.12) and (1.13) is defined as a pair of bounded functions

$$ U\left(x,t\right) = U\left(x+Ct\right) \equiv u\left(\xi \right),\kern0.61em V\left(x,t\right) = V\left(x+Ct\right) \equiv v\left(\xi \right), $$

where $C>0$ is a velocity of propagation.

Let’s now replace the capital letters in (1.9) with small letters, reduce p and q via

$p=\left(u+v\right)/{k}_1,\kern0.36em q=u-{k}_2,\kern0.24em {k}_1\ne 0$.

Take into the consideration that ${u}_t=C{u}_{\xi },\kern0.24em {u}_x={u}_{\xi };\;{v}_t=C{v}_{\xi },\kern0.24em {v}_x={v}_{\xi };\kern0.36em {u}_{xx}={u}_{\xi \xi }={v}_{\xi }/C$ and put $w={v}_{\xi },{w}_{\xi }={v}_{\xi \xi }$ we get the wave system of system (1.12) in the form

$$ \begin{array}{rcl}{u}_{\xi }&=&v/C\\ {}{v}_{\xi }&=&w\\ {}{D}_P{w}_{\xi }&=&(eC-({D}_P+{D}_Q{k}_1)/C)w-\varPhi (u,v)\end{array} $$

(1.14)

where $\varPhi \left(U,V\right)=f(U) + V\left({g}_1(U)+VG\left(U,V\right)\right)$ and functions f(u), g ₁(u) , G(u, v) are given by (1.11).

The wave system of (1.13) takes the form

$$ \begin{array}{l}{u}_{\xi }=v/C,\\ {}\left(eC-{D}_Q{k}_1\right)/C\Big){v}_{\xi }=\varPhi \left(u,v\right)\end{array} $$

(1.15)

System (1.15) contains the factor $1/\alpha \equiv \left(eC-{D}_Q{k}_1\right)/C\Big)$ which we assumed to be non-zero.

Behaviors of system (1.15) depend on the sign of parameter α. For $\alpha >0$ there exists a parameter domain containing point $M\left(e=1,{k}_1=0,{k}_2=1\right)$, where the vector field defined by system (1.15) is topologically orbitally equivalent to those defined by local system (1.1). It realizes the bifurcation of co-dimension 4 with symmetry (“spiral case”) [11]. For $\alpha <0$ parameter point $M\left(e=1,{k}_1=0,{k}_2=1\right)$ is also the point that corresponds to the bifurcation of co-dimension four with symmetry but as a “saddle case.” So, behaviors of system (1.15) for $\alpha >0$ are different from those for $\alpha <0$ (see details in [17]).