The Dynamics of Networks of Identical Theta Neurons

2.1 Derivation of Equations

2.2 Infinite N, Equally-Spaced Constants

2.2.1 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.

2.2.2 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.

2.3 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.

2.3.1 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.

2.3.2 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 2π.

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.

2.4 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.

2.5 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.

3.1 Infinite N, Equally Spaced Constants

For the case of inhibitory coupling, fixed points of type (i) shown in Fig. 3(a) gain another negative eigenvalue (as for excitatory coupling). The focus fixed point in Fig. 3(b) now has one stable direction and two unstable ones. The continuum of periodic orbits seen for \(\eta>0\) mentioned in Sect. 2.2.2 is now replaced by a single stable periodic orbit with \(\rho=1\). The frequency of this orbit increases from zero as η does, as shown in Fig. 11.

3.2 Finite N, Equally Spaced Constants

As was found in the case of instantaneous synapses, solutions with \(\rho=1\) are unaffected by this change. However, for \(\kappa=1\), the stable fixed point that exists for \(\eta >0\) and \(N=\infty\) now has small amplitude oscillations, as discussed in Sect. 2.3.1. The size of these oscillations decays exponentially with N, as was seen in Fig. 5. The only difference with the case discussed in Sect. 2.3.1 is that this low-amplitude periodic orbit is stable, whereas the one discussed in Sect. 2.3.1 was neutrally stable.

3.3 Finite N, Unequally Spaced Constants

Here we distribute the constants \(\psi_{k}\) as in Sect. 2.4 and investigate the effects of varying q on the amplitude of the oscillations in ρ of the stable periodic orbit that exists for \(\kappa=1,\eta=1,\tau=1\). The results are shown in Fig. 12. For \(q=1\), we saw previously that the amplitude of oscillations decays exponentially to zero as N increases. However, for \(q<1\), the amplitude is always finite and has a limiting value as \(N\rightarrow\infty\). As \(q\rightarrow0\), the amplitude increases and the value of N becomes less relevant. This can be seen by examining the coefficients \(C_{n}\) (13). For \(q=0\), the first \(N/2\) of \(\psi_{k}\) equal \(\pi/2\), while the last \(N/2\) equal \(-\pi/2\). Inserting this into (13), we find that \(C_{n}=0\) for n odd, and \(C_{2n}=(-1)^{n}\), independent of N.

3.4 Summary

In summary, adding synaptic dynamics of the form examined in this section destroys the reversibility of (21) and thus the non-generic behaviour of solutions of that equation. In terms of attractors, (29)–(30) show what one would expect from a self-coupled network. The only effect of having finite N is to add small fluctuations to the stable splay state.

However, in the full network (1)–(2) there can still be continuous families of attracting periodic orbits near the splay state of (29)–(30) which differ in their initial conditions, and therefore their \(\{\psi_{k}\}\). The way to understand this is to imagine picking \(\{\theta_{i}(0)\}\). Setting \(\rho=\varPsi=\varPhi=0\) gives \(\{\psi_{k}\}\), via (4), which are now fixed. (These \(\{\psi_{k}\}\) will typically not satisfy (8).) We can integrate (5)–(7) and (28) with these \(\{\psi_{k}\}\) and essentially arbitrary \(\rho(0),\varPsi (0),\varPhi(0)\) and \(I(0)\), and since κ and η are both positive, this system will have only one attractor. On this attractor the dynamics will be determined by \(\{\psi_{k}\}\). Hence different initial conditions of the \(\{\theta_{i}(0)\}\) give different attractors. These orbits have \(N-2\) Floquet multipliers of 1 and three which are less than 1 in magnitude. These three are associated with the stability of the splay state fixed point of (29)–(30).

4.1 Gap Junction Coupling

Here we consider a network of all-to-all gap junctionally coupled theta neurons. Laing [25] showed, using the equivalence of a theta neuron and a quadratic integrate-and-fire neuron [35], that a network of N identical gap junctionally coupled theta neurons can be written as follows:

$$ \frac{d\theta_{j}}{dt}=1-\cos{\theta_{j}}-g\sin{\theta_{j}}+(1+ \cos {\theta_{j}}) (I+gQ), $$

(31)

where g is the strength of coupling, I is a constant input, and

$$ Q=\frac{1}{N}\sum_{k=1}^{N} \frac{\sin{\theta_{k}}}{1+\cos{\theta _{k}}+\epsilon}, $$

(32)

where \(0<\epsilon\ll1\). We can write (31) as

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

(33)

where \(\omega=1+I+gQ\) and \(H=g+i(I+gQ-1)\). Thus the solutions of (31) can be described by the three ODEs (5)–(7) with the above definitions of ω and H. Extending the results of Laing [25], we find that

$$ Q=\sum_{m=1}^{\infty}b_{m} \gamma_{m} z^{m}+c.c., $$

(34)

where \(z=\rho e^{i\varPhi}\), “c.c.” indicates the complex conjugate of the previous term,

$$ b_{m}=\frac{i(r^{m+1}-r^{m-1})}{2(r+1+\epsilon)} $$

(35)

\(r\equiv\sqrt{2\epsilon+\epsilon^{2}}-1-\epsilon\) and

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

(36)

For \(N=\infty\) and equally-spaced \(\psi_{k}\), one can show [28] that all \(\gamma_{m}=1\) and thus (5)–(6) are sufficient to describe this system. Written in complex form, (5)–(6) are

$$ \frac{dz}{dt}=i(1+I+gQ)z+ \bigl[g+i(I+gQ-1) \bigr]/2- \bigl[g-i(I+gQ-1) \bigr]z^{2}/2. $$

(37)

Note that unlike (21)–(22), equation (37) with (34) is not invariant under \((z,t)\mapsto(\bar{z},-t)\) if \(g>0\). In a similar way to the synaptically coupled network, (37) has two fixed points with \(\rho=1\) for \(I<0\) (these correspond to all neuron states being equal). From (5)–(6), these satisfy

$$ 0=I+gQ+1+(I+gQ-1)\cos{\varPhi}-g\sin{\varPhi}, $$

(38)

which could also be derived directly from (31). These fixed points are shown in Fig. 13(b).

For large and negative I, the fixed points with \(\rho=1\) are a source and a sink, and there exists another unphysical saddle fixed point with \(\rho>1\) (Fig. 13(a)). As I is increased, the unphysical solution is involved in a transcritical bifurcation with the source fixed point having \(\rho=1\), and it enters the unit circle. As I is increased through zero, the synchronous fixed points are destroyed in a saddle-node on invariant circle (SNIC) bifurcation, leading to synchronous oscillations with \(\rho=1\), with a source fixed point inside the unit circle.

The only attractors for this system have \(\rho=1\), and for these it is clear that all \(\gamma_{m}=1\) for finite N and any \(\psi_{k}\), and thus their dynamics will still be described by (37) in this case. Solutions with \(\rho<1\) (none of which are attracting) will, however, be described by (5)–(7), and their dynamics will depend on N and the distribution of \(\psi_{k}\).

4.2 Conductance Dynamics

For excitatory coupling (\(V_{\mathrm{syn}}=2,\kappa=1\)), the fixed points are qualitatively the same as in Fig. 1, although the pair with \(\rho\neq 0\) do not have \(\varPhi=0\). This pair (analogous to those in panel (b) of Fig. 1) are a stable focus and a saddle, just as in Sect. 3.1, leading to a region of bistability. For inhibitory coupling (\(V_{\mathrm{syn}}=-2,\kappa=1\)), the fixed points are qualitatively the same as in Fig. 3; although, again, the fixed point for \(\eta>0\) does not have \(\varPhi=0\), and it is a stable focus.

We note that this model does not have the reversibility of that in Sect. 2, and thus none of the non-generic behaviour seen there. The analysis of this model for finite N should be similar to that in Sect. 3, but we do not present the results here.

4.3 Winfree Model

The Winfree model of N pulse-coupled oscillators dates from 1967 [37–39] and is written for identical oscillators as

$$ \frac{d\theta_{i}}{dt}=\varOmega+\varepsilon\frac{Q(\theta_{i})}{N}\sum _{j=1}^{N} P(\theta_{j}), $$

(44)

where we choose \(Q(\theta)=\sin{\beta}-\sin{(\theta+\beta)}\) and \(P(\theta)=a_{n}(1+\cos{\theta})^{n}\) where \(a_{n}\) is a constant such that \(\int_{0}^{2\pi}P(\theta)\,d\theta=2\pi\). The function Q is the phase response curve of an oscillator, which can be measured experimentally or determined from a model neuron [40], and \(P(\theta)\) is the pulsatile signal sent by a neuron whose state is θ. We can write (44) as

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

(45)

where \(\omega=\varOmega+\varepsilon\sigma h\) and \(H=\varepsilon e^{-i\beta}h\) and

$$ h=\frac{1}{N}\sum_{j=1}^{N} P( \theta_{j}). $$

(46)

Thus the solutions of (44) can be described by the three ODEs (5)–(7) with the above definitions of ω and H. To be concrete, choose \(n=2\), in which case

$$ h=1+2(z\gamma+\bar{z}\bar{\gamma})/3+ \bigl(z^{2}\gamma_{2}+ \bar{z}^{2}\bar {\gamma_{2}} \bigr)/6, $$

(47)

where \(z=\rho e^{i\varPhi}\) and γ and \(\gamma_{2}\) are as given in (11)–(12). As for the network of theta neurons, in the case of \(N=\infty\) and equally-spaced \(\psi_{k}\), \(\gamma=\gamma_{2}=1\) and (7) decouples from (5)–(6), which are, for this system,

$$\begin{aligned} \frac{d\rho}{dt} & =\varepsilon h \biggl(\frac{1-\rho^{2}}{2} \biggr)\cos{(\varPhi+ \beta)}, \end{aligned}$$

(48)

$$\begin{aligned} \frac{d\varPhi}{dt} & = 1+\varepsilon h \biggl[\sin{\beta}- \biggl( \frac {1+\rho^{2}}{2\rho} \biggr)\sin{(\varPhi+\beta)} \biggr], \end{aligned}$$

(49)

where we have rescaled time so that \(\varOmega=1\), and \(h=1+(4\rho /3)\cos{\varPhi}+(\rho^{2}/3)\times \cos{(2\varPhi)}\). Equations (48)–(49) are equations (12a) and (12b) in [37] once identical oscillators are considered. Equations (48)–(49) have two types of fixed point: (i) one for which \(\varPhi=\pi/2-\beta\), with ρ satisfying

$$ 0=1+\varepsilon \bigl[1+(4\rho/3)\sin{\beta}- \bigl(\rho^{2}/3 \bigr) \cos{(2\beta )} \bigr] \bigl[\sin{\beta}- \bigl(1+\rho^{2} \bigr)/(2 \rho) \bigr] $$

(50)

and (ii) up to two for which \(\rho=1\) (locked states), satisfying

$$ 0=1+\varepsilon \bigl[1+(4/3)\cos{\varPhi}+(1/3)\cos{(2\varPhi)} \bigr] \bigl[\sin{ \beta }-\sin{(\varPhi+\beta)} \bigr]. $$

(51)

These fixed points and their stability are shown in Fig. 14 as a function of ε for \(\beta=0.1\). For small ε, the only attractor is a periodic orbit with \(\rho=1\) corresponding to synchronous oscillations. As ε is increased, there is a saddle-node on invariant circle (SNIC) bifurcation destroying the periodic orbit and leading to the creation of type (ii) fixed points, one of which is stable. As ε is increased further, the type (i) fixed point undergoes a transcritical bifurcation with the saddle fixed point on \(\rho=1\) and leaves the unit circle, thereby becoming unphysical. This sequence of bifurcations happens for all \(0\leq\beta<\pi/2\). Note the similarity between this scenario and that for the gap junction coupled neurons in Sect. 4.1: both have a SNIC bifurcation on the circle \(\rho =1\), with the saddle later becoming a source as another fixed point leaves the unit circle.

As was the case in Sect. 4.1, since the only attractors have \(\rho=1\), these will persist unchanged for finite N and any \(\psi_{k}\), as γ and \(\gamma_{2}\) will both equal 1 in this case.

4.4 QIF Neurons

We implemented (52) just as in [41] but with \(N=1000\) and identical neurons. Setting \(I_{0}=0.2, J=0.1\) and using three different values of Φ and ρ to initialise \(V_{k}\), we obtained the results in Fig. 15(a). (Both the mean voltage and firing rate were smoothed by convolving with a Gaussian of standard deviation 0.07 time units.) We see three out of a continuous family of periodic orbits. They match very closely numerical simulations of (58)–(59) (not shown) even though (58)–(59) are only valid in the limit \(N\rightarrow\infty\). Conversely, if we choose \(V_{k}(0)\) not using (61), for example, taking them randomly from a unit normal distribution, we obtain the results in Fig. 15(b). The solution is quasiperiodic and clearly not described by a planar system. To understand this, consider the transformation (4) with \(\theta_{k}(0)=2\tan^{-1}(V_{k}(0))\). For large N, it is generally impossible to choose \(\varPhi(0),\varPsi(0)\) and \(\rho(0)\) such that \(\psi_{k}\) are uniformly distributed. Thus the system must be described by three coupled equations of the form (5)–(7) which, even in the limit \(N\rightarrow\infty\), will not reduce to a planar system due to the non-uniformity of the distribution of \(\psi_{k}\).

Note that while (52) is synaptically driven by the instantaneous rate r, the network (1) is driven by the average of the pulse-like functions in (2). However, replacing \((1-\cos{\theta})^{2}\) in (2) by \(a_{n}(1-\cos{\theta})^{n}\), where \(a_{n}\) is chosen so that the integral of this function over \([0,2\pi]\) is independent of n, and taking the limit \(n\rightarrow\infty\), one can obtain a drive equal (up to a scale factor) to the rate given in terms of delta functions in (53) [24, 36].