1. Introduction
Many physical entities such as neurons and lasers can be modelled as oscillators [Reference Ermentrout and Terman5, Reference Pikovsky, Rosenblum and Kurths19]. Coupling them together results in a network of coupled oscillators. The effect of one oscillator on others in a network may be delayed due to, for example, the finite speed of light, or of action potentials propagating along axons [Reference Ermentrout and Ko3, Reference Ermentrout and Terman5].
One of the simplest model oscillators is the theta neuron [Reference Ermentrout and Kopell4], which is the normal form of the saddle-node-on-invariant-circle (SNIC) bifurcation [Reference Gutkin7]. A theta neuron has a single parameter, I, which can be chosen so that the neuron is either excitable or active (periodically firing). It has the advantage that its state can be found explicitly as a function of time for constant I [Reference Laing and Krauskopf14]. In a previous paper [Reference Laing and Krauskopf14] we considered a single theta neuron with delayed self-coupling (an autapse [Reference Seung, Lee, Reis and Tank21]) in the form of a Dirac delta function of time. The action of a delta function on a theta neuron can be easily calculated, so we were able to analytically describe periodic solutions of this model and determine their stability, giving a complete description of the types of periodic solutions, where they occur in parameter space and their stability.
More recently we considered a pair of theta neurons, each coupled to the other through delayed delta functions [Reference Laing and Krauskopf15]. We considered the case of excitable neurons and found two types of periodic solutions: those for which the neurons were perfectly synchronous, and those for which the neurons were half a period out of phase with one another. Extending the analysis in [Reference Laing and Krauskopf14] we derived explicit expressions for the existence and stability of both types of solutions. We also described symmetry-broken solutions and analytically determined their stability. We found disconnected branches of solutions, all of which lose stability when the period of a solution as a function of delay is at a minimum.
This paper considers a pair of theta neurons, each coupled to the other through delayed delta functions, but when the uncoupled neurons are active. We perform an analysis similar to that in [Reference Laing and Krauskopf15], finding two continuous branches of periodic solutions, one for which the neurons are perfectly synchronous, and another for which they alternate firing. These branches undergo symmetry-breaking bifurcations whenever the period as a function of delay is either a maximum or a minimum. The model is presented in Section 2, synchronous solutions are studied in Section 3, and alternating ones in Section 4. Symmetry-broken solutions are studied in Section 5. We consider the case of smooth feedback in Section 6 and conclude in Section 7.
2. Model
We first consider a single theta neuron [Reference Ermentrout and Kopell4] governed by
$$ \begin{align} \frac{d\theta}{dt}=1-\cos{\theta}+(1+\cos{\theta})I, \end{align} $$
where
$\theta \in [0,2\pi )$
and I is a positive constant. The solution of equation (2.1) is
$$ \begin{align*} \theta(t)=2\tan^{-1}\bigg[\sqrt{I}\tan\bigg(\sqrt{I}t+\tan^{-1}\bigg(\frac{\tan[\theta(0)/2]}{\sqrt{I}}\bigg)\bigg)\bigg]. \end{align*} $$
In what follows we set
$I=1$
, and thus a single theta neuron satisfies
$d\theta /dt=2$
and
$\theta (t)=\theta (0)+2t$
. (While this may seem to be a drastic assumption, if
$I\neq 1$
, by letting
$\tan {(\theta /2)}=\sqrt {I}\tan {(\phi /2)}$
we find that
$d\phi /dt=2$
[Reference Monteforte and Wolf18].)
In this paper, we consider a pair of such neurons coupled with each other via delayed Dirac delta functions, described by
$$ \begin{align} \frac{d\theta_1}{dt} & =1-\cos{\theta_1}+(1+\cos{\theta_1})\bigg(1+\kappa\sum_{i:t-\tau<s_i<t} \delta(t-s_i-\tau)\bigg), \end{align} $$
$$ \begin{align} \frac{d\theta_2}{dt} & =1-\cos{\theta_2}+(1+\cos{\theta_2})\bigg(1+\kappa\sum_{i:t-\tau<t_i<t} \delta(t-t_i-\tau)\bigg), \end{align} $$
where
$\tau $
is the (constant) delay and firing times in the past of neuron 1 can be written
$\{\ldots , t_{-3},t_{-2},t_{-1},t_0\}$
, and those of neuron 2 can be written
$\{\ldots , s_{-3},s_{-2},s_{-1},s_0\}$
. The constant
$\kappa $
is the strength of coupling between the neurons. The influence of the delta function is to increment
$\theta $
using
$$ \begin{align} \tan{(\theta^+/2)}=\tan{(\theta^-/2)}+\kappa, \end{align} $$
where
$\theta ^-$
is the value of
$\theta $
before the delta function acts and
$\theta ^+$
is the value after [Reference Laing and Krauskopf14]. Such a network with
$I=-1$
(that is, when both neurons are excitable rather than active) and
$0<\kappa $
was considered in [Reference Laing and Krauskopf15].
Example solutions of (2.2)–(2.3) are shown in Figure 1. In this paper, we focus on solutions of the form shown: either both neurons are perfectly synchronous, or they are half a period out of phase with one another. Since between the times at which a delta function acts we have
$d\theta /dt=2$
, and we know the effect of the delta function, (2.4), we can analytically construct solutions such as those in Figure 1 and determine their stability. In Section 3 we consider synchronous solutions and in Section 4 we consider alternating solutions.
3. Synchronous solutions
We first consider periodic solutions of (2.2)–(2.3) for which the neurons are perfectly synchronous, as shown in the top row of Figure 1. The influence of one neuron on the other is thus the same as that of the neuron on itself. The existence of such solutions is governed by the same equation that governs the behaviour of a single neuron delay-coupled to itself [Reference Laing and Krauskopf14].
3.1. Existence
As shown in [Reference Laing and Krauskopf14], perfectly synchronous periodic solutions of (2.2)–(2.3) with period T satisfy
$$ \begin{align} (n+1)T=\tau+\frac{\pi}{2}-\tan^{-1}\bigg[\kappa+\tan\bigg(\tau-nT+\frac{\pi}{2}\bigg)\bigg], \end{align} $$
where
$\tan ^{-1}$
is the arctangent function and n is the number of past firing times in the interval
$(-\tau ,0)$
, assuming that a neuron has just fired at time
$t=0$
. The primary branch of solutions, corresponding to
$n=0$
, is given explicitly by
$$ \begin{align} T(\tau)=\tau+\frac{\pi}{2}-\tan^{-1}\bigg[\kappa+\tan\bigg(\tau+\frac{\pi}{2}\bigg)\bigg] \end{align} $$
for
$0\leq \tau \leq \pi $
, while secondary branches are given parametrically, using the reappearance of periodic solutions in delay differential equations with fixed delays [Reference Yanchuk and Perlikowski25], as
$$ \begin{align} (\tau,T)=(s+nT(s),T(s)), \end{align} $$
where
$0\leq s\leq \pi $
. Several branches of such solutions are shown in blue in Figure 2.
3.2. Stability
We now derive the stability of a synchronous periodic solution. Suppose neuron 1 last fired at time
$t_0$
and neuron 2 last fired at
$s_0$
where
$s_0\approx t_0$
. The most distant past firing of neuron 1 in
$(t_0-\tau ,t_0)$
is
$t_{-n}$
and the most distant past firing of neuron 2 in
$(s_0-\tau ,s_0)$
is
$s_{-n}$
.
Blue: synchronous periodic solutions (solid stable, dashed unstable). The nth branch goes from
$(n\pi ,\pi )$
to
$((n+1)\pi ,\pi )$
. Red: alternating periodic solutions (solid stable, dashed unstable). The nth branch goes from
$((n-1/2)\pi ,\pi )$
to
$((n+1/2)\pi ,\pi )$
. Black: symmetry-broken periodic solutions (all unstable, except the branch at
$\tau =0$
which is neutrally stable). The filled circles indicate saddle-node bifurcations.
$\kappa =2$
.
For neuron 1, from
$t_0$
we wait
$\tau -(t_0-s_{-n})$
at which point neuron 1 has its phase incremented due to a past firing of neuron 2. Before the reset,
$\theta _1$
equals
$$ \begin{align*} \theta_1^-=\pi+2(\tau-(t_0-s_{-n})), \end{align*} $$
and after reset it is
$\theta _1^+$
, where
$$ \begin{align*} \tan{(\theta_1^+/2)}=\tan{(\theta_1^-/2)}+\kappa. \end{align*} $$
Neuron 1 will then fire after a further time
$\Delta _1$
where
$$ \begin{align*} \Delta_1=\frac{\pi-\theta_1^+}{2}. \end{align*} $$
Thus,
$$ \begin{align} t_1 & =t_0+\tau-(t_0-s_{-n})+\Delta_1 \nonumber \\ & =\tau+s_{-n}+\pi/2-\tan^{-1}[\kappa+\tan(\pi/2+\tau-(t_0-s_{-n}))]. \end{align} $$
Similarly, for neuron 2, from time
$s_0$
we wait
$\tau -(s_0-t_{-n})$
until neuron 2 has its phase incremented as a result of the past firing of neuron 1. Before the reset
$\theta _2$
equals
$$ \begin{align*} \theta_2^-=\pi+2(\tau-(s_0-t_{-n})), \end{align*} $$
and after the reset it equals
$\theta _2^+$
, where
$$ \begin{align*} \tan{(\theta_2^+/2)}=\tan{(\theta_2^-/2)}+\kappa. \end{align*} $$
Neuron 2 will then fire after a further time
$\Delta _2$
, where
$$ \begin{align*} \Delta_2=\frac{\pi-\theta_2^+}{2}. \end{align*} $$
So,
$$ \begin{align} s_1 & =s_0+\tau-(s_0-t_{-n})+\Delta_2 \nonumber \\ & =\tau+t_{-n}+\pi/2-\tan^{-1}[\kappa+\tan(\pi/2+\tau-(s_0-t_{-n}))]. \end{align} $$
Equations (3.4) and (3.5) give
$t_1$
and
$s_1$
in terms of previous firing times, but in general we have
$$ \begin{align} s_{i+1} & =\tau+t_{i-n}+\pi/2-\tan^{-1}[\kappa-\cot(\tau-(s_i-t_{i-n}))] , \end{align} $$
$$ \begin{align} t_{i+1} & =\tau+s_{i-n}+\pi/2-\tan^{-1}[\kappa-\cot(\tau-(t_i-s_{i-n}))], \end{align} $$
where we used
$\tan {(\pi /2+x)}=-\cot {x}$
. We write (3.6)–(3.7) as
$$ \begin{align*} F(s_{i+1},t_{i-n},s_i) & = 0, \\ G(t_{i+1},s_{i-n},t_i) & = 0. \end{align*} $$
To find the stability of a solution, we perturb
$t_i\to t_i+\eta _i$
and
$s_i\to s_i+\mu _i$
. Then for linear order we have
$$ \begin{align*} \frac{\partial F}{\partial s_{i+1}}\mu_{i+1}+\frac{\partial F}{\partial t_{i-n}}\eta_{i-n}+\frac{\partial F}{\partial s_{i}}\mu_{i} & =0, \\[5pt] \frac{\partial G}{\partial t_{i+1}}\eta_{i+1}+\frac{\partial G}{\partial s_{i-n}}\mu_{i-n}+\frac{\partial G}{\partial t_{i}}\eta_{i} & =0, \end{align*} $$
which, after evaluating the partial derivatives at a periodic solution with period T, we write as
$$ \begin{align} -\mu_{i+1}+(1-\gamma)\eta_{i-n}+\gamma\mu_{i}=0, \end{align} $$
$$ \begin{align} -\eta_{i+1}+(1-\gamma)\mu_{i-n}+\gamma\eta_{i}=0, \end{align} $$
where
$$ \begin{align} \gamma=\frac{\csc^2{(\tau-nT)}}{1+[\kappa-\cot{(\tau-nT)}]^2}. \end{align} $$
This is the same quantity as was found in [Reference Laing and Krauskopf14], when studying the stability of a periodic solution of a self-coupled theta neuron. Assuming solutions of the linear equations (3.8)–(3.9) of the form
$\mu _i=A\lambda ^i$
and
$\eta _i=B\lambda ^i$
for some constants A and B, as in [Reference Laing and Krauskopf15], we obtain the characteristic equation for the multipliers,
$\lambda $
, as follows:
$$ \begin{align*} F_a(\lambda)\equiv \lambda^{2n+2}-2\gamma\lambda^{2n+1}+\gamma^2\lambda^{2n}-(1-\gamma)^2=0. \end{align*} $$
This is the same equation as was found in [Reference Laing and Krauskopf15], where two excitable neurons were studied, the only difference being the definition of
$\gamma $
. The magnitudes of the roots of
$F_a(\lambda )$
determine the stability of the perfectly synchronous periodic solution. If all roots have
$|\lambda |\leq 1$
the periodic solution is not unstable, but if one or more roots have
$|\lambda |> 1$
the periodic solution is unstable.
We first consider the case
$n=0$
. Then
$F_a(\lambda )=(\lambda -1)(\lambda +1-2\gamma )$
. The root
$\lambda =1$
reflects the invariance of the system to uniform time translation, and since
$0<\gamma $
the only instability that can occur is when
$\gamma =1$
. This point corresponds to
$dT/d\tau =0$
on the primary branch. To see that this is the case, differentiating (3.2) with respect to
$\tau $
, we find that
$dT/d\tau =1-\gamma $
where
$\gamma $
is given by (3.10) with
$n=0$
. Thus
$dT/d\tau =0$
when
$\gamma =1$
.
Summarizing the results in [Reference Laing and Krauskopf15] for
$0<n$
, we find that such a synchronous solution undergoes two types of bifurcations, one when
$dT/d\tau =0$
and the other at a saddle-node bifurcation (that is, when the curve of period, T, as a function of delay,
$\tau $
, is either vertical or horizontal) on each branch, indexed by n.
3.3. Branches of solutions
Plotting branches of solutions as given by equations (3.2)–(3.3) for
$\kappa =2$
, and indicating their stability, we obtain the blue curve in Figure 2. Note that these curves are the same as shown in [Reference Laing and Krauskopf14, Figure 7], but their stability is different, due to the possibility of losing stability to a solution which is not synchronous. These symmetry broken states are shown in black in Figure 2, and they are analysed in Section 5.1. Note that if on an unstable section of a branch there are two saddle-node bifurcations (marked with filled circles in Figure 2) there are two unstable multipliers between the bifurcations. Stable solutions lose stability in symmetry-breaking bifurcations when
$dT/d\tau =0$
, and between a symmetry-breaking and a saddle-node bifurcation a solution has one unstable multiplier.
4. Alternating solutions
We now consider solutions for which the neurons take turns firing, half a period out of phase with one another, as shown in the bottom row of Figure 1.
4.1. Existence
As shown in [Reference Laing and Krauskopf15], the existence of alternating solutions of (2.2)–(2.3) is given by (3.1) under the replacement of
$\tau $
by
$\tau +T/2$
:
$$ \begin{align} (n+1/2)T=\tau+\frac{\pi}{2}-\tan^{-1}\bigg[\kappa+\tan\bigg(\tau-(n-1/2)T+\frac{\pi}{2}\bigg)\bigg]. \end{align} $$
The meaning of n in (4.1) is that if neuron 1 fires at time 0, there are n past firing times of neuron 2 in
$(-\tau ,0)$
; n could be zero.
4.2. Stability
Performing a similar analysis as in Section 3.2 or in [Reference Laing and Krauskopf15], we obtain the firing time maps, valid when the oscillators are approximately half a period out of phase:
$$ \begin{align} t_{i+1} & = \tau+s_{i+1-n}+\pi/2-\tan^{-1}{[\kappa+\tan{(\pi/2+\tau-(t_i-s_{i+1-n}))}]} , \end{align} $$
$$ \begin{align} s_{i+1} & = \tau+t_{i-n}+\pi/2-\tan^{-1}{[\kappa+\tan{(\pi/2+\tau-(s_i-t_{i-n}))}]}\qquad \end{align} $$
for
$i=0,1,2,\ldots .$
We want to linearize around an alternating periodic solution of (4.2)–(4.3). To do that, write (4.2)–(4.3) as
$$ \begin{align*} R(t_{i+1},s_{i-n+1},t_i) & = 0, \\ S(s_{i+1},t_{i-n},s_i) & = 0, \end{align*} $$
then perturb the firing times and assume that these perturbations either grow or decay exponentially with index. The calculations are similar to those in Section 3.2, and we obtain the characteristic equation governing the stability of these solutions as
$$ \begin{align} F_b(\lambda)\equiv\lambda^{2n+1}-2\gamma\lambda^{2n}+\gamma^2\lambda^{2n-1}-(1-\gamma)^2=0, \end{align} $$
where
$$ \begin{align} \gamma= \frac{\csc^2{(\tau-(n-1/2)T)}}{1+[\kappa-\cot{(\tau-(n-1/2)T)}]^2}. \end{align} $$
This characteristic equation was found in [Reference Laing and Krauskopf15] for the case of two excitable neurons, but in that paper
$\gamma $
referred to a different quantity, not that in (4.5). Using the results in [Reference Laing and Krauskopf15] for the roots of (4.4), we have that the alternating periodic solution with
$n=0$
is stable for
$0<\gamma <1$
and unstable for
$1<\gamma $
. For
$0<n$
each branch of alternating periodic solutions undergoes two bifurcations when the curve of T as a function of
$\tau $
is either vertical or horizontal, just as for the synchronous solutions. Branches of these solutions are shown in red in Figure 2, with stability indicated. Saddle-node bifurcations are also shown.
5. Symmetry-broken solutions
As mentioned, both types of solutions analysed above undergo bifurcations when
$dT/d\tau =0$
. These are symmetry-breaking bifurcations and we now analyse the resulting solutions.
5.1. Symmetry-breaking from synchronous solutions
We start with (3.6)–(3.7) and break the symmetry, so that
$s_i-t_{i-n}=(n-\phi )T$
and
$t_i-s_{i-n}=(n+\phi )T$
; thus
${\phi =0}$
corresponds to the perfectly synchronous case. Substituting these into (3.6)–(3.7), we obtain equations for the existence of such states:
$$ \begin{align} \tan{(\pi/2+\tau-(n+1-\phi)T)} & = \kappa+\tan{(\pi/2+\tau-(n-\phi)T)} , \end{align} $$
$$ \begin{align} \tan{(\pi/2+\tau-(n+1+\phi)T)} & = \kappa+\tan{(\pi/2+\tau-(n+\phi)T)}. \end{align} $$
Using the identity
$\tan {a}-\tan {b}=\sin {(a-b)}/(\cos {a}\cos {b})$
first on (5.1) and then on (5.2), and the fact that cosine is an even function, we find that solutions of (5.1)–(5.2) satisfy
$T=2\tau /(2n+1)$
. In this case both (5.1) and (5.2) reduce to
$$ \begin{align} \cot{((1/2-\phi)T)} = \kappa-\cot{((1/2+\phi)T)}. \end{align} $$
For fixed
$\kappa $
, solutions of (5.3) lie on a curve in
$(T,\phi )$
space, as shown in Figure 3. The curves terminate at
$\phi =\pm 1/2$
, and these values correspond to alternating solutions. When
$\phi =\pm 1/2$
we see from (5.3) that
$T=\pi $
, independent of
$\kappa $
. When
$\phi =0$
we have
$T=2\cot ^{-1}{(\kappa /2)}$
. Thus the symmetry-broken solutions lie on the lines
${T=2\tau /(2n+1)}$
, where
$(2n+1)\cot ^{-1}{(\kappa /2)}\leq \tau \leq (n+1/2)\pi $
, and are plotted in black in Figure 2 emanating from each minimum on the curve of synchronous solutions (shown in blue). They each terminate at a maximum on the curve of alternating solutions (shown in red). Note that only every second of the black curves shown in Figure 2 is described by this analysis; the other curves are analysed in Section 5.2. The stability of these types of solutions can be calculated as in [Reference Laing and Krauskopf15] and they are all unstable.
Solutions of (5.3), describing symmetry-broken solutions, for
$\kappa =4,2,1$
(left to right).
5.2. Symmetry-breaking from alternating solutions
5.2.1.
$\tau =0$
solutions
We see from Figure 2 that a symmetric alternating solution exists for
$\tau =0$
. But a whole family of asymmetric solutions also exist, shown with the vertical black line at
$\tau =0$
in Figure 2. We now analyse them.
Between firing times the flow is given by
$d\theta _1/dt=2$
and
$d\theta _2/dt=2$
. Assume that
$\theta _2$
has just fired (that is,
$\theta _2=\pi $
) and
$\theta _1=\alpha $
where
$0<\alpha <\pi $
. Both
$\theta _1$
and
$\theta _2$
will increase until
$\theta _1=\pi $
, which takes a time
$\Delta _1=(\pi -\alpha )/2$
, at which point
$\theta _2=2\pi -\alpha $
. The phase
$\theta _2$
is then incremented to
$\theta _2^+=2\tan ^{-1}{(\kappa +\tan {(\pi -\alpha /2)})}$
. Both phases then continue to increase until
$\theta _2=\pi $
, which takes a further time
${\Delta _2=(\pi -\theta _2^+)/2}$
, at which point
$\theta _1=\pi +2\Delta _2=2\pi -\theta _2^+$
. The phase
$\theta _1$
is then incremented to
${\theta _1^+=2\tan ^{-1}{(\kappa +\tan {(\pi -\theta _2^+/2)})}}$
. For this process to describe a periodic solution we need
$\theta _1^+=\alpha $
, which is true for all
$0<\alpha <\pi $
. (A similar calculation can be done for
${\pi <\alpha <2\pi }$
.) Thus there is a continuum of such periodic solutions.
The period of such a solution is
$T=\Delta _1+\Delta _2$
and so we can write
${\Delta _1=(1/2+\phi )T}$
and
$\Delta _2{\kern-1pt}={\kern-1pt}(1/2-\phi )T$
for some
$-1/2{\kern-1pt}<{\kern-1pt}\phi {\kern-1pt}<{\kern-1pt}1/2$
, where
$\phi {\kern-1pt}={\kern-1pt}0$
corresponds to the symmetric alternating solution. We find that
$\cot {(\Delta _1)}{\kern-1pt}={\kern-2pt}\tan {(\alpha /2)}$
and
${\cot {(\Delta _2)}{\kern-1pt}={\kern-1pt}\kappa {\kern-1pt}-{\kern-2pt}\tan {(\alpha /2)}}$
and thus
$\cot {(\Delta _2)}=\kappa -\cot {(\Delta _1)}$
, or
$$ \begin{align} \cot{((1/2-\phi)T)} = \kappa-\cot{((1/2+\phi)T)}, \end{align} $$
which is identical to (5.3), and whose solutions are shown in Figure 3. This family of asymmetric solutions lie on the T-axis with
$2\cot ^{-1}{(\kappa /2)}<T\leq \pi $
, and are shown in black in Figure 2. These solutions are neutrally stable, as there is a continuum of them.
5.2.2.
$\tau>0$
solutions
The solutions in the previous section exist for
$\tau =0$
. Using the reappearance of solutions of delay differential equations, we see that a solution with a given
$\phi $
and T which satisfies equation (5.4) is also a periodic solution with the same
$\phi $
and T when the delay equals a multiple of T. Thus, these symmetry-broken solutions lie on the lines
$T=\tau /n$
with
$2n\cot ^{-1}{(\kappa /2)}<\tau \leq n\pi $
; these are shown black in Figure 2. These lines leave minima on the curves of alternating solutions (shown in red) when
$\phi =0$
, and terminate at maxima on curves of synchronous solutions (shown in blue) when
$\phi =\pm 1/2$
. The stability of these solutions can be determined using calculations similar to those in [Reference Laing and Krauskopf15], and they are unstable.
Periodic solutions of (6.1)–(6.2). Blue: synchronous solutions. Red: alternating solutions. Solid: stable. Dashed: unstable. The symmetry-broken solutions (all unstable) are shown in black. Filled circles show the points at which the number of unstable Floquet multipliers of a solution has changed from one to two; these are saddle-node bifurcations.
$m=5,\kappa =2$
.
6. Smooth feedback
We now consider the case of smooth feedback, to see whether the results for Dirac delta function coupling persist. The equations we study are
$$ \begin{align} \frac{d\theta_1}{dt} & =1-\cos{\theta_1}+(1+\cos{\theta_1})\{1+\kappa P[\theta_2(t-\tau)]\} , \end{align} $$
$$ \begin{align} \frac{d\theta_2}{dt} & =1-\cos{\theta_2}+(1+\cos{\theta_2})\{1+\kappa P[\theta_1(t-\tau)]\}, \end{align} $$
where
$$ \begin{align*} P(\theta)=a_m(1-\cos{\theta})^m, \end{align*} $$
with
$a_m=2^m(m!)^2/(2m)!$
, is a pulsatile function centred at
$\theta =\pi $
with
$\int _0^{2\pi }P(\theta )d\theta =2\pi $
for all m. Increasing m makes this function “sharper”, and in the limit
$m\to \infty $
we have
$P(\theta )=2\pi \delta (\theta -\pi )$
where
$\delta $
is the Dirac delta function [Reference Laing13].
We set
$m=5$
and find branches of synchronous and alternating solutions using DDE-BIFTOOL [Reference Sieber, Engelborghs, Luzyanina, Samaey and Roose22]. They are plotted in Figure 4, as are the symmetry-broken solutions, with stability indicated. We find perfect qualitative agreement with the results shown in Figure 2, obtained for delta function coupling, showing the robustness of our results.
7. Discussion
We exactly described periodic solutions that occur in a pair of delay-coupled active theta neurons, and analytically found their stability. Our work is an extension of that in [Reference Laing and Krauskopf15] where a pair of excitable theta neurons were studied. The results are similar, in that symmetry-breaking instabilities were found where
$dT/d\tau =0$
. To obtain periodic solutions for excitable systems we needed excitatory coupling, that is,
$0<\kappa $
. The results in this paper also had
$0<\kappa $
, but that is not necessary to see periodic solutions in networks of active neurons. The analysis performed here is equally valid for inhibitory coupling (
$\kappa <0$
), the main difference being that all solutions will have periods greater than or equal to
$\pi $
(the period of an uncoupled neuron) as inhibition can only slow down oscillations.
We now briefly discuss similar work by others. A number of authors have considered delay-coupled phase oscillators which rotate at a constant speed when uncoupled, as we do. However, some choose the interactions between oscillators to be smooth, depending on sinusoidal functions of phase differences, for example [Reference Earl and Strogatz2, Reference Ermentrout and Ko3, Reference Schuster and Wagner20, Reference Yeung and Strogatz26]. Others consider uniformly rotating oscillators with delayed delta function coupling [Reference Ernst, Pawelzik and Geisel6, Reference Klinshov and Nekorkin12, Reference Li, Lin and Efstathiou17, Reference Woodman and Canavier24], but none have used the update rule (2.4) specific to a theta neuron with pulsatile current input. As an example, Klinshov et al. [Reference Klinshov, Lücken, Shchapin, Nekorkin and Yanchuk9] study a model containing a phase resetting curve
$Z(\theta ^-)=\theta ^+-\theta ^-$
, where
$\theta ^-$
is the value of
$\theta $
before the delta function acts and
$\theta ^+$
is the value after. For the update rule (2.4) we have
$$ \begin{align*} Z(\theta)=2\tan^{-1}{[\tan{(\theta/2)}+\kappa]}-\theta. \end{align*} $$
One can show that
$-1<Z'(\theta )$
, so neither a single self-coupled theta neuron nor a pair of them as considered here can undergo a “multijitter” bifurcation of the type seen in [Reference Klinshov, Lücken, Shchapin, Nekorkin and Yanchuk9–Reference Klinshov, Lücken, Yanchuk, Nekorkin, Edelman, Macau and Sanjuan11].
We note that a number of authors (including the authors in [Reference Laing and Longtin16]) write
$d\theta /dt=[\cdots ]+f(\theta )\delta (t-\tau )$
to indicate that
$\theta $
is incremented by the amount
$f(\theta )$
at
$t=\tau $
. However, this interpretation of the impulsive differential equation is incorrect [Reference Catllá, Schaeffer, Witelski, Monson and Lin1, Reference Klinshov, Lücken and Feketa8]. Alternating and synchronous periodic solutions were found in a pair of delay-coupled FitzHugh–Nagumo systems [Reference Song and Xu23]; however, this work and [Reference Laing and Krauskopf15] are the most comprehensive studies of this phenomena so far, aided by the analytical solutions of the models under study.
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