In this section we consider instantaneous synapses in the sense that the synaptic input to a neuron, in the form of a current, depends on the present state of the neurons which are connected to it.

### Derivation of Equations

Suppose we have a network of *N* (\(3< N\)) identical theta neurons, all-to-all coupled by instantaneous synapses. The dynamics is described by [24–27]

$$\begin{aligned} \frac{d\theta_{k}}{dt} =1-\cos{\theta_{k}}+(1+\cos{\theta_{k}}) (\eta+\kappa I ) \end{aligned}$$

(1)

for \(k=1,2,\ldots, N\), where

$$ I=\frac{1}{N}\sum_{j=1}^{N}(1-\cos{ \theta_{j}})^{2}, $$

(2)

*κ* is the strength of coupling (which could be positive or negative), and *η* is the input current to all neurons when uncoupled. The *j*th term in the sum (2) represents the pulse of current emitted by the *j*th neuron as it fires, i.e. \(\theta_{j}\) increases through *π*. The function \((1-\cos{\theta})^{2}\) is non-zero except when \(\theta =0\), and thus this form of coupling may be regarded as non-physical, but the pulse can be localised more around \(\theta=\pi\) by increasing the second power in (2) as will be discussed below. Note that each neuron is actually coupled to itself, but this term is only one out of *N* in the sum (2), so will be negligible for large *N*.

We can write (1) as

$$ \frac{d\theta_{k}}{dt}=\omega+\operatorname{Im} \bigl[\mathrm{He}^{-i\theta_{k}} \bigr], $$

(3)

where \(\omega=\eta+\kappa I+1\) and \(H=i(\eta+\kappa I-1)\). Note that *ω* is real and *H* is imaginary. The WS ansatz [9, 28] states that there is a transformation

$$ \tan{ \biggl(\frac{\theta_{k}(t)-\varPhi(t)}{2} \biggr)}=\frac{1-\rho (t)}{1+\rho(t)}\tan{ \biggl( \frac{\psi_{k}-\varPsi(t)}{2} \biggr)};\quad k=1,2,\ldots, N, $$

(4)

giving almost any solution of (1)–(2) (\(\theta_{k}\)) in terms of *N* constants (\(\{\psi_{k}\}, k=1,2,\ldots, N\)) and three variables (\(\rho,\varPhi\) and *Ψ*), where these variables satisfy the ODEs

$$\begin{aligned} \frac{d\rho}{dt} & = \frac{1-\rho^{2}}{2}\operatorname{Re} \bigl[\mathrm{He}^{-i\varPhi} \bigr], \end{aligned}$$

(5)

$$\begin{aligned} \frac{d\varPhi}{dt} & = \omega+\frac{1+\rho^{2}}{2\rho}\operatorname {Im} \bigl[\mathrm{He}^{-i\varPhi} \bigr], \end{aligned}$$

(6)

$$\begin{aligned} \frac{d\varPsi}{dt} & = \frac{1-\rho^{2}}{2\rho}\operatorname{Im} \bigl[\mathrm{He}^{-i\varPhi } \bigr]. \end{aligned}$$

(7)

Thus, while it may seem possible that (for a given initial condition) a solution of (1)–(2) can explore the full *N*-dimensional phase space described by the *N* values of \(\theta_{k}\), the solution is actually constrained to lie on a three-dimensional manifold with coordinates \(\rho,\varPhi\) and *Ψ*. The dynamics on this manifold will depend on the values of the constants \(\{\psi_{k}\}\).

There are *N* variables \(\{\theta_{k}\}\), *N* constants \(\{\psi_{k}\}\) and three variables \(\rho,\varPhi\) and *Ψ*, and thus one needs to specify three constraints so that there is a unique relationship between \(\{\theta_{i}\}\) and \(\{\psi_{k},\rho,\varPhi,\varPsi\}\), to determine initial conditions, for example. One way to do this is to set \(\rho(0)=\varPhi(0)=\varPsi(0)=0\), so that \(\{\psi_{k}\}=\theta _{k}(0)\). Then integrate (5)–(7) with \(\rho(0)=\varPhi (0)=\varPsi(0)=0\) and use the solutions \(\rho(t),\varPhi(t)\) and \(\varPsi(t)\) to reconstruct \(\{\theta_{k}(t)\}\) using (4). Here, the constraints are on \(\rho(0),\varPhi(0)\) and \(\varPsi(0)\). Another set of constraints often taken is [28]

$$ \sum_{k=1}^{N} e^{i\psi_{k}}=0;\qquad \operatorname{Re} \Biggl[\sum_{k=1}^{N} e^{2i\psi _{k}} \Biggr]=0. $$

(8)

Given the *N* initial values \(\theta_{k}(0)\), one can usually uniquely determine the values of \(\rho(0),\varPhi(0)\) and \(\varPsi(0)\) and \(\{\psi_{k}\}\) such that both (4) and (8) hold [9]. “Usually” refers to solutions for which fewer than half of the neurons have exactly the same state. This rules out full synchrony, which is often a state of interest; however, we can understand the fully-synchronous state by simply considering the behaviour of one self-coupled neuron. We will use (8) unless specified otherwise.

In order to use (5)–(7), we need to write *I* (and thus *ω* and *H*) in terms of the new variables and the constants \(\{\psi_{k}\}\). Now [28]

$$\begin{aligned} I & =3/2-\frac{1}{N}\sum_{j=1}^{N} \bigl(e^{i\theta_{j}}+e^{-i\theta _{j}} \bigr)+\frac{1}{4N}\sum _{j=1}^{N} \bigl[ \bigl(e^{i\theta _{j}} \bigr)^{2}+ \bigl(e^{-i\theta_{j}} \bigr)^{2} \bigr] \end{aligned}$$

(9)

$$\begin{aligned} & = 3/2-(z\gamma+\bar{z}\bar{\gamma})+ \bigl(z^{2} \gamma_{2}+ \bar{z}^{2}\bar {\gamma_{2}} \bigr)/4, \end{aligned}$$

(10)

where overbar indicates complex conjugate, \(z=\rho e^{i\varPhi}\),

$$ \gamma=\frac{1}{N}\sum_{k=1}^{N} \frac{1+ \vert z \vert ^{-2}\bar{z}e^{i(\psi_{k}+\varPhi-\varPsi)}}{1+\bar{z}e^{i(\psi_{k}+\varPhi -\varPsi)}} =\frac{1}{N\rho}\sum_{k=1}^{N} \frac{\rho+e^{i(\psi_{k}-\varPsi )}}{1+\rho e^{i(\psi_{k}-\varPsi)}} $$

(11)

and

$$ \gamma_{2}=\frac{1}{N}\sum_{k=1}^{N} \biggl(\frac{1+ \vert z \vert ^{-2}\bar{z}e^{i(\psi_{k}+\varPhi-\varPsi)}}{1+\bar{z}e^{i(\psi _{k}+\varPhi-\varPsi)}} \biggr)^{2} =\frac{1}{N\rho^{2}}\sum _{k=1}^{N} \biggl(\frac{\rho+e^{i(\psi_{k}-\varPsi )}}{1+\rho e^{i(\psi_{k}-\varPsi)}} \biggr)^{2}. $$

(12)

We define

$$ C_{n}=\frac{1}{N}\sum_{k=1}^{N} e^{in\psi_{k}} $$

(13)

and see that \(C_{0}=1\), and from (8), \(C_{1}=0\). Using a series expansion of \([1+\rho e^{i(\psi_{k}-\varPsi)}]^{-1}\), we can write in the general case

$$ \gamma=1+ \bigl(1-1/\rho^{2} \bigr)\sum_{n=2}^{\infty}C_{n} \bigl(-\rho e^{-i\varPsi} \bigr)^{n}. $$

(14)

For the special case of \(\psi_{k}=2\pi k/N\), i.e. evenly spaced \(\psi_{k}\), \(C_{n}=0\) except when *n* is a multiple of *N*, when it equals 1. Then (14) is a geometric series and

$$ \gamma=1+\frac{(1-1/\rho^{2})(-\rho e^{-i\varPsi})^{N}}{1-(-\rho e^{-i\varPsi})^{N}} $$

(15)

(note that this expression is given incorrectly in [17, 21]).

In the general case,

$$ \gamma_{2}=\sum_{n=0}^{\infty}(n+1) \bigl(-\rho e^{-i\varPsi} \bigr)^{n} \biggl[C_{n}+ \frac{2e^{-i\varPsi}}{\rho}C_{n+1}+\frac{e^{-2i\varPsi }}{\rho^{2}}C_{n+2} \biggr]. $$

(16)

For evenly-spaced \(\psi_{k}\),

$$\begin{aligned} \gamma_{2}= {}& \sum_{k=0}^{\infty}(Nk+1) \bigl(-\rho e^{-i\varPsi} \bigr)^{Nk} +\sum _{k=1}^{\infty}Nk\frac{2e^{-i\varPsi}}{\rho} \bigl(-\rho e^{-i\varPsi } \bigr)^{Nk-1} \\ &{} +\sum_{k=1}^{\infty}(Nk-1)\frac{e^{-2i\varPsi}}{\rho^{2}} \bigl(-\rho e^{-i\varPsi} \bigr)^{Nk-2} \end{aligned}$$

(17)

$$\begin{aligned} ={}& N\sum_{k=1}^{\infty}k \bigl(-\rho e^{-i\varPsi} \bigr)^{Nk}+\sum_{k=0}^{\infty}\bigl(-\rho e^{-i\varPsi} \bigr)^{Nk} -\frac{2N}{\rho^{2}}\sum _{k=1}^{\infty}k \bigl(-\rho e^{-i\varPsi} \bigr)^{Nk} \\ &{} + \frac{N}{\rho^{4}}\sum_{k=1}^{\infty}k \bigl(-\rho e^{-i\varPsi } \bigr)^{Nk}-\frac{1}{\rho^{4}}\sum _{k=1}^{\infty}\bigl(-\rho e^{-i\varPsi} \bigr)^{Nk} \end{aligned}$$

(18)

$$\begin{aligned} = {}& 1+\frac{(1-1/\rho^{4})(-\rho e^{-i\varPsi})^{N}}{1-(-\rho e^{-i\varPsi })^{N}}+N\frac{(1-1/\rho^{2})^{2}(-\rho e^{-i\varPsi})^{N}}{[1-(-\rho e^{-i\varPsi })^{N}]^{2}}. \end{aligned}$$

(19)

Irrespective of \(\psi_{k}\), if \(\rho=1\) then \(\gamma=\gamma_{2}=1\). If \(\rho<1\) and \(\{\psi_{k}\}\) are uniformly distributed, then as \(N\rightarrow\infty\) we see that \(\gamma\rightarrow1\) and \(\gamma_{2}\rightarrow1\). In these cases (\(\gamma=\gamma_{2}=1\)) *I* becomes independent of *Ψ* and (5) and (6) decouple from (7), i.e. the dynamics becomes two-dimensional.

An alternative description of the dynamics of (1)–(2) when \(N=\infty\) is given by writing the continuity equation governing the evolution of the probability density of *θ*s, \(p(\theta,t)\) [26]. One can then use the Ott/Antonsen (OA) ansatz [29, 30] to reduce the dynamics of this evolution equation to a single complex equation for the evolution of the order parameter \(z\equiv\int_{0}^{2\pi}p(\theta,t)e^{i\theta}\,d\theta\):

$$ \frac{dz}{dt}=i(\eta+kI) (1+z)^{2}/2-i(1-z)^{2}/2=i \omega z+H/2-\bar {H}z^{2}/2, $$

(20)

where *ω* and *H* are as above and \(I=3/2-(z+\bar{z})+(z^{2}+\bar{z}^{2})/4\). Substituting \(z=\rho e^{i\varPhi}\) into (20) and taking real and imaginary parts, we recover (5) and (6). Thus the OA ansatz corresponds to a special case of the WS ansatz: when \(N=\infty\) and the constants \(\{\psi_{k}\}\) are uniformly spread over \([0,2\pi ]\) [28]. Note that while the OA ansatz gives dynamics on an invariant manifold in the space of all \(p(\theta,t)\), if the neurons are identical, the manifold is not attracting, and thus the full dynamics is not given by (20) and must be described using the WS ansatz [17, 28].

### Infinite *N*, Equally-Spaced Constants

We now consider the case of \(N=\infty\) with uniform density of \(\psi_{k}\). Thus we are interested in solutions of (20), written as

$$ \frac{dz}{dt}=i(\eta+\kappa I+1)z+i(\eta+\kappa I-1) \bigl(1+z^{2} \bigr)/2, $$

(21)

where

$$ I=3/2-(z+\bar{z})+ \bigl(z^{2}+\bar{z}^{2} \bigr)/4. $$

(22)

Note that (21)–(22) are invariant under \((z,t)\mapsto(\bar{z},-t)\), i.e. simultaneous reflection in the real axis and reversal of time. This has a significant effect on the possible dynamics.

#### Excitatory Coupling

First consider \(\kappa=1\). From setting \(d\rho/dt=0\) in (5) and recalling that *H* is imaginary, we find two sorts of fixed points of (21)–(22): (i) those with \(\rho=1\) and (ii) those with \(\varPhi=0\). From (6), those with \(\rho=1\) satisfy

$$ 0=\omega+\operatorname{Im} \bigl[\mathrm{He}^{-i\varPhi} \bigr]=\eta+\kappa I+1+(\eta+\kappa I-1) \cos{\varPhi}, $$

(23)

where

$$ I=3/2-2\cos{\varPhi}+\cos{(2\varPhi)}/2. $$

(24)

These solutions are plotted in Fig. 1(a). Note that these solutions can be found directly from (1)–(2). For identical neurons, \(\rho=1\) corresponds to full locking, so all \(\theta_{i}\) are equal to *Φ* and a simple trigonometric identity gives (24) from (2).

Solutions of type (ii) with \(\varPhi=0\) have \(z=\rho\). From (6), they satisfy

$$ 0=\omega+\frac{1+\rho^{2}}{2\rho}\operatorname{Im}[H]=\eta+\kappa I+1+\frac {1+\rho^{2}}{2\rho}(\eta+ \kappa I-1), $$

(25)

where \(I=3/2-2\rho+\rho^{2}/2\). These solutions (restricted to \(-1\leq \rho\leq1\) for physical reasons) are shown in Fig. 1(b). Two solutions are created in a saddle-node bifurcation as *η* is increased (at a negative value of *η*). One is a saddle whose eigenvalues sum to zero, and the other is a focus with purely imaginary eigenvalues; these properties result from the reversibility of the system [31]. These fixed points correspond to *splay states* in which all neurons follow the same trajectory but are equally displaced from one another in time such that average quantities such as *z* are constant [9, 32, 33]. Note that the solutions shown in Fig. 1(a) collide with the saddle solution in Fig. 1(b) at \((\eta ,z)=(0,1)\).

A selection of solutions and the fixed points are shown in Fig. 2(a) for \(\eta=-0.5\). We see that for these parameter values, initial conditions either tend to the stable fixed point in the lower half plane (i.e. quiescence), or (if they are in the region enclosed by the homoclinic orbit to the saddle fixed point) follow one of a continuous family of periodic orbits. For \(\eta>0\), the only remaining fixed point is the focus, and the phase space (\(\rho\leq1\)) is filled with a continuous family of periodic orbits (see Fig. 2(b)), again, as a result of the system’s reversibility.

#### Inhibitory Coupling

Now consider \(\kappa=-0.5\). As with excitatory coupling, there are fixed points with \(\rho=1\), given by (23)–(24) and shown in Fig. 3(a). For \(\eta>0\), there is also a focus fixed point (a splay state), as shown in Fig. 3(b). This fixed point is surrounded by a continuum of periodic orbits, as in Fig. 2(b). (Note that for strong inhibition, i.e. *κ* large and negative, the situation shown in Fig. 3(a) can become more complex, with multiple stable fixed points. This is due to the finite width of the pulses in (2), and the region of multistability disappears as the pulses are made narrower.)

In summary, the dynamics of (21)–(22) is non-generic due to their reversible nature, which can lead to the existence of continuous families of neutrally-stable periodic orbits.

### Finite *N*, Equally-Spaced Constants

We now consider the case of finite *N* but with equally-spaced \(\{\psi _{k}\}\). Thus we consider (5)–(7) where *I* depends on *Ψ* via *γ* and \(\gamma_{2}\). First we point out the reversibility of this system under the transformation \((\rho,\varPhi,\varPsi,t)\mapsto(\rho,-\varPhi,-\varPsi,-t)\). This transformation interchanges the products *zγ* and *z̄γ̄*, and \(z^{2}\gamma_{2}\) and its conjugate. This leaves *I* (and thus *ω* and *H*) unchanged. Keeping in mind that *H* is imaginary, we see that this transformation leaves (5)–(7) unchanged, i.e. they are reversible.

#### Fixed Points

The fixed points of type (i) with \(\rho=1\) persist, independent of *N*, since these solutions have \(\gamma=\gamma_{2}=1\). The values of *Φ* are given by solving (23)–(24). However, these fixed points have arbitrary values of *Ψ* since \(d\varPsi/dt=0\). Regarding the fixed points of type (ii) that exist for \(N=\infty\) as analysed in Sect. 2.2.1, they had constant and generically non-zero \(d\varPsi/dt\). Thus, for finite *N*, we expect these to appear as time-dependent orbits, with the amplitudes of fluctuations in *ρ* and *Φ* going to zero as \(N\rightarrow\infty\). To understand this, assume to a first approximation that *ρ* is constant. Then *γ* and \(\gamma_{2}\) will have *N* periods of oscillation as *Ψ* goes through one period of oscillation. Thus \(I,\omega\) and *H* will all have *N* oscillations in one period of *Ψ* and so will *ρ* and *Φ*. Thus the fixed points of type (ii) which exist for infinite *N* will appear for finite *N* as quasiperiodic orbits in which *ρ* and *Φ* undergo *N* oscillations for each full rotation in *Ψ*. An example for \(N=4\) is shown in Fig. 4, where *Ψ* decreases from *π* to −*π*.

The amplitude of this type of periodic orbit goes to zero exponentially as \(N\rightarrow\infty\); see Fig. 5. Even though *N* is physically an integer, the expressions for *γ* and \(\gamma_{2}\) do not require it to be so. Thus we can, for example, continue the saddle-node bifurcation seen in Fig. 1(b) as a function of the continuous parameter *N*. The result is shown in Fig. 6, where we also show interpolated values at integer *N*. Interestingly, while the curve oscillates, the values at integer *N* are monotonic.

In terms of the original system (1)–(2), the periodic orbit shown in Fig. 4 corresponds to a periodic orbit with all Floquet multipliers having magnitude 1, i.e. completely neutrally stable. Two of the multipliers are a complex conjugate pair corresponding to the rotation seen in Fig. 2, while the remaining \(N-2\) are equal to 1.

#### Other Orbits

Consider the continuous family of periodic orbits which exist in the infinite-*N* case for \(\kappa=1\) and *η* sufficiently positive (see Fig. 2(b)). These persist as what seems to be a continuous family of quasiperiodic orbits. Some are shown in Fig. 7 where we plot the value of *z* when \(\alpha\equiv\varPhi-\varPsi\) increases through a multiple of 2*π*.

Now consider the case \(\kappa=-0.5\). The dynamics in this case seems much more complex. An example is shown in Fig. 8 for \(\eta=0.6\). For some initial conditions, the dynamics seems quasiperiodic, while for others the orbits appear to be chaotic (as indicated by a positive Lyapunov exponent, not shown). This mixture of quasiperiodic and chaotic behaviour has been previously observed in reversible systems [34] including a resistively-loaded series array of Josephson junctions also studied using the WS ansatz [22]. The overall trend for this system is that the dynamics becomes more regular as *η* is increased. We leave the investigation of this dynamics for a future publication.

### Finite *N*, Non-Uniform \(\psi_{k}\)

Now consider non-uniformly spaced constants \(\psi_{k}\). We follow [28] and distribute these values along two arcs each of length *qπ*, where \(0< q\leq1\) is a parameter. Choosing *N* to be even, we have \(\psi_{k}=(1-q)\pi/2-q\pi/N+2\pi qk/N\) for \(k=1,2,\ldots, N/2\) and \(\psi_{k}=(3-q)\pi/2-q\pi/N+2\pi qk/N\) for \(k=N/2+1,2,\ldots, N\). For \(q=1\), this is a uniform distribution, and as \(q\rightarrow0\) the distribution tends to the two points \(\pm\pi/2\). The saddle-node bifurcation shown in Fig. 1(b) moves as a function *q*, as shown in Fig. 9. Varying *q* for \(\kappa<0\) also gives a variety of different dynamics, as shown in Fig. 10. Here we see a mixture of quasiperiodic and more complex behaviour, as seen in Fig. 8.

### Summary

In summary, for instantaneous synapses of the form studied here, there may be continuous families of periodic orbits (for drive \(\eta>0\) and for some \(\eta<0\) if \(\kappa>0\)) in the infinite-*N* uniformly distributed \(\psi_{k}\) case. This is due to the reversibility of (21)–(22). For finite *N*, some of these orbits become quasiperiodic, with some initial conditions showing either quasiperiodic or more complex types of behaviour. This type of network can be thought of as having two sources of degeneracy: even if the constants \(\{\psi_{k}\}\) are fixed, depending on parameters, there may be continuous families of neutrally stable periodic or quasiperiodic orbits. Choosing different initial conditions for the WS variables \(\rho,\varPhi\) and *Ψ* selects between these orbits. Secondly, even for fixed initial conditions of the WS variables, there are many continuous families of orbits that can be obtained by varying \(\{\psi_{k}\}\).

Note that if the system is bistable (for excitatory coupling and sufficiently small negative drive *η*), then even for fixed initial values of \(\rho,\varPsi\) and *Φ*, changing either *N* (for evenly spaced constants \(\psi_{k}\)) or the distribution of \(\psi_{k}\) (for fixed *N*) can lead to very different outcomes, as these changes can move the system from one basin of attraction to another.