Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2010 / B / 6

The common vector field $F$ represents an identical reference cell. The small cell-specific term $ϵG_i$ represents heterogeneity of cellular properties, such as differing ion-current parameters or intrinsic oscillation rates. It is not external random noise: the displayed model is deterministic.

The Kronecker delta factors restrict coupling to the first state component, the membrane potential. Substitution gives a coupling current in the voltage equation proportional to $ϵ{\sum }_j{d}_{ij}(u_j^1-u_i^1)$, while the other state components have no direct intercellular coupling. This is the linear voltage-difference response of gap junctions. Gating and chemical variables can still change indirectly through their own coupled intracellular dynamics. The zero diagonal removes self-coupling, which would contribute zero anyway. Physical reciprocal gap-junction coupling is often symmetric, but zero diagonal alone does not impose reciprocity; that distinction matters in the final part.

With $\sigma =\omega (ϵ)t$, $\tau =ϵt$, the multiscale chain rule gives $d/dt=(1+ϵ{\Omega }_1+\cdots ){\partial }_{\sigma }+ϵ{\partial }_{\tau }$. Expanding $F(u_{i,0}+ϵu_{i,1})=F(u_{i,0})+ϵDF(u_{i,0})u_{i,1}+\cdots$ gives at order oneAt order $ϵ$, moving the linearized reference dynamics to the left givesThe component sum runs over the $m$ state variables; the source uses $n$ in its first displayed sum and $m$ later. All functions for the solvability calculation are periodic in $\sigma$ with period one.

Differentiating the autonomous cycle equation $U^{\prime }=F(U)$ gives $U^{\prime \prime }=DF(U)U^{\prime }$, so $U^{\prime }$ lies in the periodic null space of $\hat{L}$. For a hyperbolic attracting limit cycle, its tangent direction has Floquet multiplier one and every transverse multiplier has modulus below one. Hence it is the only periodic direction, proving the required one-dimensional null space.

Strictly, mere nonlinear stability does not ensure this spectral assertion. In polar coordinates $\dot{r}=-(r-1{)}^3$, $\dot{\phi }=2\pi$ has an attracting unit-period cycle but neutral radial linearization at $r=1$, supplying a second periodic solution of the variational equation. Thus the intended phase reduction assumes the usual simple unit Floquet multiplier, or equivalently hyperbolic transverse attraction; this assumption is made explicit here.

Use the period-one inner product $⟨Y,V⟩={\int }_0^{1}Y(\sigma )\cdot V(\sigma )d\sigma$. Periodic integration by parts gives ${\hat{L}}^{\ast }Y=-Y^{\prime }-DF(U{)}^{\mathsf{T}}Y$. Let $Y$ span this adjoint null space and normalize ${\int }_0^{1}Y\cdot U^{\prime }d\sigma =1$. For oscillator $i$, write $s=\sigma +{\theta }_i(\tau )$. Then ${\partial }_{\tau }{u}_{i,0}={\theta }_i^{\prime }{U}^{\prime }(s)$ and ${\partial }_{\sigma }{u}_{i,0}=U^{\prime }(s)$.

The Fredholm solvability condition makes the forcing in the order-$ϵ$ equation orthogonal to $Y(s)$. Its intrinsic and frequency terms integrate to ${\xi }_i-{\Omega }_1-{\theta }_i^{\prime }$, whereVoltage-only coupling contributesThus withthe complete phase equation isPeriodicity makes these formulae independent of the absolute initial phase. The adjoint function is a phase response function; it projects a small perturbation of each state component onto its accumulated phase change. The subtraction of $h(0)$ ensures zero coupling between identical phases, as required by the original voltage differences. The source's reference to its equation (1) is read as the uncoupled cycle equation, numbered (3) on the actual page.

At a locked state choose the rotating frame with ${\theta }_i^{\prime }=0$. For small phase differences, Taylor expansion gives $h({\theta }_j-{\theta }_i)-h(0)=h^{\prime }(0)({\theta }_j-{\theta }_i)+O(|{\theta }_j-{\theta }_i{|}^2)$. Let the graph Laplacian have $L_{ii}={\sum }_j{d}_{ij}$ and $L_{ij}=-d_{ij}$ for $i\ne j$. The leading locked equations areIf coupling is balanced, ${\sum }_j{d}_{ij}={\sum }_j{d}_{ji}$ for every $i$, so $1^{\mathsf{T}}L=0$. Summing over cells cancels all linear coupling terms and givesIn particular symmetric reciprocal gap-junction conductances satisfy this condition. To leading order the collective physical frequency is $1+ϵN^{-1}{\sum }_i{\xi }_i$.

The arithmetic-mean conclusion requires this extra balance assumption. The printed condition $d_{ii}=0$ alone is insufficient. For a connected directed coupling network let ${\pi }^{\mathsf{T}}L=0$, normalized by ${\pi }^{\mathsf{T}}1=1$. The valid weighted collective oscillator frequency result isFor two cells with $d_{12}=p>0$, $d_{21}=q>0$, one has $\pi =(q,p)/(p+q)$ and ${\Omega }_1=(q{\xi }_1+p{\xi }_2)/(p+q)$ at linear order. For example $p=1,q=2,{\xi }_1=0,{\xi }_2=\delta$ gives ${\Omega }_1=\delta /3$, rather than $\delta /2$, while the locked phase difference is $O(\delta )$ when $h^{\prime }(0)\ne 0$. This supplies a counterexample with arbitrarily small phase differences. It also identifies exactly why reciprocal coupling recovers the requested mean: the left null space is then spanned by the uniform vector.