We design and analyze the dynamics of a large network of theta neurons, which are idealized type I neurons. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global, via pulselike synapses of adjustable sharpness. Using recently developed analytical methods, we identify all possible asymptotic states that can be exhibited by a mean field variable that captures the network's macroscopic state. These consist of two equilibrium states that reflect partial synchronization in the network and a limit cycle state in which the degree of network synchronization oscillates in time. Our approach also permits a complete bifurcation analysis, which we carry out with respect to parameters that capture the degree of excitability of the neurons, the heterogeneity in the population, and the coupling strength (which can be excitatory or inhibitory). We find that the network typically tends toward the two macroscopic equilibrium states when the neuron's intrinsic dynamics and the network interactions reinforce one another. In contrast, the limit cycle state, bifurcations, and multistability tend to occur when there is competition among these network features. Finally, we show that our results are exhibited by finite network realizations of reasonable size.
The cortex of the brain is structured into a complex network of many functional neural assemblies (Sherrington, 1906; Hebb, 1949; Harris, 2005). Each assembly typically encompasses a large number of interacting neurons with various dynamical characteristics. Within this network, communication among the neural assemblies generally involves macroscopic signals that arise from the collective behavior of the constituent neurons in each assembly. It has been proposed that functional behavior arises from these interactions, and perceptual representations in the brain are believed to be encoded by the macroscopic spatiotemporal patterns that emerge in these networks of assemblies.
In modeling the brain, a microscopic description of individual neurons and their interactions is important. However, a model for the macroscopic dynamical behavior of large assemblies of neurons is essential for understanding the brain's emergent, collective behavior (Peretto, 1984; Sompolinsky, 1988; Kanamaru & Masatoshi, 2003). In this letter, we construct such a model using the canonical theta neuron of Ermentrout and Kopell (Ermentrout & Kopell, 1986; Ermentrout, 1996). We construct a large heterogeneous network, containing both excitable and spiking neurons, that is globally coupled via smooth pulselike synapses. Then, using the methods of Ott and Antonsen (2008, 2009) and Marvel, Mirollo, and Strogatz (2009) (see also Pikovsky & Rosenblum, 2008, 2011), we derive a low-dimensional “reduced” dynamical system that exhibits asymptotic behavior that is coinciding with that of the macroscopic mean field of the network. We use this reduced system to classify all the asymptotic macroscopic configurations that the network can exhibit.
We show that our network exhibits three fundamental collective states: a partially synchronous rest state (PSR), a partially synchronous spiking state (PSS), and a collective periodic wave state (CPW). In the PSR state, most neurons remain at rest but are excitable, and the macroscopic mean field sits on a stable equilibrium. In the PSS state, the mean field is also on a stable equilibrium, but typically, most individual neurons spike regularly. These states are similar to states that have been called asynchronous (Abbott & van Vreeswijk, 1993; Hansel & Mato, 2001, 2003). We find that they are typically encountered in cooperative networks in which the internal dynamics of the neurons and the interneuronal network interactions reinforce each other. In other parameter regions where the internal dynamics and network interaction are in competition, a collective periodic state, the CPW state, can occur. In the CPW state, the phases of the neurons transiently cluster, and the degree of network coherence waxes and wanes periodically in time in such a way that the macroscopic mean field exhibits a stable limit cycle.
These three macroscopic states can coexist and transition into each other, and here we clarify precisely how this happens using our reduced mean field equation. We obtain a complete bifurcation diagram with respect to parameters that represent the degree of excitability, heterogeneity, and the strength of coupling (both excitatory and inhibitory) within the network. Our model provides a comprehensive description for the asymptotic macroscopic behavior of a large network of heterogeneous theta neurons in the thermodynamic limit.
The remainder of this letter is organized as follows. In section 2, we describe the basic features of our theta neuron network, and in section 3, we derive the mean field reduction using the Ott-Antonsen method (Ott & Antonsen, 2008, 2009; Marvel et al., 2009). Then, in section 4, we use the reduced mean field equation to identify and describe the three possible macroscopic states: PSR, PSS, and CPW. In section 5, we provide a comprehensive bifurcation analysis for the macroscopic dynamics of the network. We summarize and discuss our results in section 6.
2. Microscopic Formulation
The brain contains a huge number of different types of neurons that feature various dynamical characteristics. Here, we construct a mathematically tractable network that encompasses three fundamental characteristics: neuronal excitability, pulselike coupling, and heterogeneity.
2.1. Neuronal Excitability: The Theta Neuron.
A typical neuron at rest begins to spike as a constant injected current exceeds a threshold. Neurons are usually classified into two types based on this behavior (Hodgkin, 1948; Ermentrout, 1996; Izhikevich, 2007). Type I neurons begin to spike at an arbitrarily slow rate, whereas type II neurons spike at a nonzero rate as soon as the threshold is exceeded. Neurophysiologically, excitatory pyramidal neurons are often of type I, and fast-spiking inhibitory interneurons are often of type II (Nowak, Azouz, & Sanchez-Vives, Gray, & McCormick, 2003; Tateno, Harsch, & Robinson, 2004).1
In this letter, we construct our network using only theta neurons. Future work will consider networks containing mixtures of type I and type II neurons.
2.2. Coupling via a Pulselike Synapse.
3. Mean Field Reduction of the Network
We now derive a reduced dynamical system that exhibits asymptotic behavior that coincides with the asymptotic behavior of the mean field z(t) of the network. Our procedure follows Ott and Antonsen (2008, 2009) and Marvel et al. (2009).
4. Macroscopic Dynamics of the Network
By analyzing the reduced mean field equation, equation 3.13, we find that there are three possible asymptotic macroscopic states for the network. Two of these, which we call the partially synchronous rest (PSR) state and the partially synchronous spiking (PSS) state, correspond to equilibria of the macroscopic mean field. The third, which we call the collective periodic wave (CPW) state, corresponds to a limit cycle of the macroscopic mean field.
4.1. Macroscopic Equilibrium States.
In the PSR state, the macroscopic mean field z(t) of the network settles onto a stable node. This state is predominantly (but not exclusively) observed when the distribution of excitability parameters is such that most neurons are in the resting regime (e.g., ), and the neurons are coupled through inhibitory synapses (k<0). Thus, most neurons are inactive, with their phase angles residing near their resting states. Nevertheless, some spiking neurons are present. These come from the tail of the distribution and have a negligible effect on the collective behavior of the network. Figure 3a shows an example of the macroscopic PSR equilibrium, with its local invariant manifolds calculated using the reduced mean field equation, equation 3.13, using , , k=−2, and n=2. (A movie showing both the macroscopic and microscopic behavior of the PSR state in Figure 3 is available in the online supplement.) Figures 3b and c show a trajectory, after the initial transient behavior has been discarded, of the macroscopic mean field z(t) calculated from a network of N=10, 000 theta neurons with the same system parameters. As expected, the slightly noisy trajectory hovers about the predicted equilibrium with fluctuations roughly on the order of .
In the PSS state, the macroscopic mean field z(t) settles onto a stable focus. The collective behavior of the infinite network is again at rest, but in this case, there is an intrinsic circulation, as the local stability of the equilibrium is given by a pair of complex eigenvalues (with negative real parts). This state occurs predominantly (but not exclusively) when most neurons inherently spike (), with the coupling being either excitatory (k>0) or weakly inhibitory (). Thus, although most neurons are active, the network is partially synchronous and organized such that phase cancellation among the neurons results in a well-defined stationary macroscopic mean.
Figure 4a shows an example of the macroscopic PSS state obtained using the reduced system, equation 3.13, with , , k=2, and n=2. (A movie showing both the macroscopic and microscopic behavior of the PSS state in Figure 4 is available in the online supplement.) As before, panels b and c show the mean field trajectory z(t) of a network of N=10, 000 neurons at the same parameter values. The reduced system once again accurately predicts the ultimate macroscopic state of the network, and finite-size network effects reveal bouts of coherent circulation about the focus.
Both the PSR and PSS states exhibit stationary behavior in the macroscopic mean field z(t) and reflect partially coherent network configurations. We emphasize here the subtle difference between them: one is a node in the macroscopic mean field and the other is a focus. This observation suggests that transient behavior in the macroscopic mean field z(t) resulting from abrupt perturbations of network parameters should reveal the difference between these two states.
Figure 5 shows time series of the macroscopic mean field z(t) for both the PSR (panels a and b) and the PSS (panels c and d) states. For the PSR state, the system starts with the following parameter set: , , k=−2, and n=2. Then, at t=500, is abruptly switched from −0.2 to −0.5. The new asymptotic state remains a PSR state (with Lyapunov exponents ), but the stable node shifts, and the macroscopic mean field z(t) converges exponentially toward the new asymptotic value. The time series from both the reduced system (see Figure 5a) and a discrete network of 10, 000 neurons (see Figure 5b) clearly demonstrates this exponential convergence.
The results from applying the same procedure to a PSS state (with , k=2, and n=2, and changing from 0.2 to 0.5) are shown in Figures 5c and 5d. In this case, the perturbed PSS state is characterized by a stable focus with a pair of stable complex eigenvalues (). Thus, the transient behavior after the parameter shift exhibits prominent oscillations that do not occur in the PSR case.
The PSR and PSS states are the expected macroscopic states when the network's intrinsic dynamics (as parameterized by the excitability parameter ) and the synaptic interactions (as characterized by the coupling k) are not in competition. The PSR state tends to occur when a majority of the neurons are intrinsically at rest () and the overall coupling is inhibitory (k<0). Conversely, the PSS state tends to occur when a majority of the neurons inherently spike () and the overall coupling is excitatory (k>0). In these cases, the internal dynamics and the network interaction reinforce each other, and the resulting macroscopic dynamics is a simple equilibrium. We show in the following section that a more complicated dynamical state can occur when the intrinsic neuronal dynamics and the network interaction compete with one another.
4.2. Macroscopic Limit Cycle State and Multistability.
In the CPW state, the macroscopic mean field settles onto a stable limit cycle, and z(t) oscillates in time. We adopt this terminology based on previous work in which a collective oscillatory state of this type, when viewed from the microscopic perspective, was described as a wave (Crawford, 1994; Ermentrout, 1998; Bressloff, 1999; Osan, Rubin, & Ermentrout, 2002). Here, we find that this state occurs when most neurons are active () and the synaptic interaction is inhibitory (k<0). The microscopic configuration of the neurons is such that the degree of coherence waxes and wanes in time as the phases of the neurons corral together and spread apart in a periodic manner. Thus, the collective oscillation reflects the interplay between the neurons’ inherent tendency to spike and the suppressive network interaction. Indeed, we show below that the occurrence of CPW states is mediated by Andronov-Hopf and homoclinic bifurcations of the mean field, and thus are emergent properties of the network. In particular, the frequency of the limit cycle for a CPW state is not simply related to the frequencies of the individual neurons. In addition, the macroscopic limit cycle takes on different shapes and sizes for different system parameters, thus indicating different microscopic wave patterns.
A particular example of the CPW state is shown in Figure 6. As before, panel a shows the attractors predicted by equation 3.13 with , , k=−9, and n=2. (A movie showing both the macroscopic and microscopic behavior of the CPW state in Figure 6 is available in the online supplement.) In this case, an attracting CPW limit cycle and a PSR node coexist. Two unstable equilibria are present as well. Panel b shows the asymptotic mean field behavior of a network of 10, 000 neurons, where separate runs with different initial conditions were used to demonstrate the coexistence of the two attractors. Once again, the reduced mean field equation gives an excellent prediction for the asymptotic temporal behavior of the full network.
For most regimes in parameter space, the macroscopic behavior of the network is found to exclusively approach just one of the above defined states; there is only a single macroscopic attractor. However, there are significant parameter regions in which the network exhibits multistability, where two or more of these macroscopic states are found to coexist. Indeed, the example shown in Figure 6 is an example of multistability in which both a stable CPW state and a stable PSR state coexist. For parameters within these multistable regions, the network approaches one of the stable macrostates depending on how the neurons in the network are configured initially. We find, based on our bifurcation analysis of the mean field equation (see section 5), that dynamical competition is a necessary ingredient for the emergence of multistability. (A more detailed analysis of the multistable state for a similar but nonautonomous network of theta neurons is reported in So et al. (2013).) (A singular situation occurs with , corresponding to a homogeneous network of neurons. Due to the high degree of symmetry present in this case, the collective behavior consists of many coexisting neutrally stable limit cycles, and the overall macroscopic dynamics can be counterintuitively more complicated than the heterogeneous case studied here. A more detailed analysis of this homogeneous case and other extensions of this work will be reported elsewhere.)
One can also entertain the notion of unstable macroscopic PSR, PSS, or CPW states, as we already mentioned. Although these are not typically observable in the collective behavior of the physical network, we will demonstrate in the next section that they play an important role in mediating the transitions among the three classes of observable (attracting) macroscopic states.
5. Bifurcation Analysis of the Macroscopic States
Having identified the three classes of attractors for the macroscopic mean field z(t), we now turn our attention to the analysis of the bifurcations that they can undergo. Specifically, we identify the bifurcations that occur as the following network parameters are varied: the neurons’ intrinsic excitability parameter , the heterogeneity parameter , and the overall coupling strength k. We consider both excitatory (k>0) and inhibitory (k<0) interaction among the neurons. The bifurcation set will be illustrated in the three-dimensional parameter space defined by , , and k, for fixed values of the synaptic sharpness parameter n. In our examples, we use n=2 and n=9, and our results suggest that the bifurcation scenarios described here are qualitatively robust with respect to n.
For a generic codimension 1 bifurcation such as the saddle node (SN) or the Andronov-Hopf (AH) bifurcation, this procedure results in two-dimensional surfaces embedded in the full five-dimensional space. We can visualize these two-dimensional bifurcation sets in the three-dimensional space defined by the network parameters , , and k. In sections 5.1 and 5.2, we examine the saddle node and the Andronov-Hopf bifurcations, respectively. We infer (and numerically verify) that homoclinic bifurcations are present as well. We describe in section 5.3 the transition between the PSR and the PSS states in which a macroscopic equilibrium changes from a node to a focus, or vice versa. We call this a node focus (NF) transition. This transition is not typically classified as a bifurcation in the traditional sense, since the stability of the equilibrium does not change, and additional states are neither created nor destroyed. Nevertheless, it is desirable to know where in parameter space this transition occurs, since the type of equilibrium (i.e., focus or node) can have macroscopic consequences, as illustrated in Figure 5. Collectively, these results lead to an understanding of the various bifurcations and transitions that occur in the attractors of the macroscopic mean field of our network.
5.1. Saddle Node Bifurcation.
The bifurcation set consists of two similar tentlike structures. The edges of the tentlike surfaces correspond to parameter values where a codimension 2 cusp bifurcation occurs. It is notable that these tentlike structures are predominantly (but not exclusively) located in regions where the internal excitability parameter and the coupling strength k are of opposite sign for both excitatory and inhibitory connectivity. This is the dynamically competitive region mentioned above. Furthermore, the similarity between the surfaces in a (for n=2) and b (for n=9) indicate the robustness of our results with respect to the synaptic sharpness parameter n.
Panel a includes two line segments—one parallel to the k axis (with ) and the other parallel to the axis (with ). Panel c is a magnification of panel a in the vicinity of the former, showing how this line segment pierces the SN surfaces. These line segments are keyed to Figures 8c and 11c, discussed below, in order to clarify which macroscopic states exist and which bifurcations occur as parameters are traversed along these lines.
Figure 8a shows a two-dimensional slice through the n=2 tent at . A typical fold structure with two saddle node curves meeting at a codimension 2 cusp point is seen. Panel b shows the one-dimensional bifurcation diagram, plotting y=Im(z) versus k, that results from following k along the line (dashed line in panel a; this is the same as the vertical line segment in Figures 7a and 7c). This diagram shows how the equilibrium solutions evolve as k increases from zero. Initially there is an attracting PSS state. This changes into a PSR state at the NF transition point indicated by the open diamond at k=0.1028. As k increases further, this PSR state gradually migrates toward higher values of y. Then a SN bifurcation and a NF transition point occur in rapid succession at k=0.9067 and k=0.9075, respectively (these points are not resolvable at the resolution shown in the figure and are therefore marked “SN/NF”). This SN bifurcation creates new stable and unstable PSR states in a separate region of state space (near y=−0.08), and at the NF point, the stable PSR changes into a stable PSS state. As k increases further, this stable PSS state persists, while the unstable PSR state migrates toward smaller values of y and collides with the coexisting stable PSR state. These annihilate each other via the SN bifurcation at k=1.1237. This sequence of events is shown schematically collapsed onto the k axis in panel c, so that it can be compared to the vertical line segment in Figures 7a and 7c. The shaded region indicates an interval in which more than one stable attractor exists. Note also that an NF transition occurs at a negative value of k (−0.5697) that is not visible in panel b.
5.2. Andronov-Hopf Bifurcation.
The result is a tube or funnel-shaped surface that opens and flattens out on one side. The funnel emanates from the regime of large inhibitory coupling and less heterogeneity with (i.e., most neurons are very close to their SNIC bifurcations), and then opens up and flattens out for increasing values of (i.e., greater dominance of spiking neurons). As in the case of the SN bifurcation, the surface occurs most prominently where there is dynamic competition within the network. However, in this case, the surface exists only where the competition is specifically between predominantly active neurons and inhibitory network interaction ( and k<0).
Figure 10a shows the two-dimensional bifurcation diagram that results from slicing through the n=2 AH and SN surfaces at k=−9. The two SN curves again meet at a cusp, and the AH curve intersects the left SN curve at a codimension 2 Bogdanov-Takens (BT) point. The dotted rectangular region shown in panel a is magnified in panel b, making it easier to see the AH curve, as well as the homoclinic (HC) bifurcation curve that also emerges from the BT point (Kuznetsov, 2004). We identified the latter curve numerically.4
To further clarify the identity of the macroscopic network states, Figure 11a shows the one-dimensional bifurcation diagram (in this case, x=Re(z) versus ) obtained by varying along the line (dashed lines in Figure 10; these are the same as the horizontal lines in Figures 7a and 9a). Here, the heavy solid lines represent stable equilibria. The lower equilibrium branch corresponds to the PSR state, and it persists until it collides with an unstable PSR state in a saddle node bifurcation. Moving along the upper stable equilibrium with decreasing , the network exhibits the PSS state before encountering the AH bifurcation, which is supercritical. At this point, the equilibrium loses stability, and an attracting limit cycle emerges: the CPW state. The amplitude of this limit cycle subsequently increases until it collides with the unstable (uPSR) equilibrium in a homoclinic bifurcation.
Figure 11b shows a magnification of the vicinity of the SN/NF point in panel a, showing both the SN and NF points distinctly. This SN point corresponds to the left SN curve in Figure 10a and in this case, leads to the creation of two unstable PSR states.
5.3. PSR to PSS Transition.
In Figures 8 and 11, we indicated that the stable equilibria depicted there make transitions between the PSR and PSS states. These transitions occur when the equilibrium changes from a node (PSR state; real eigenvalues) to a focus (PSS state; complex eigenvalues), or vice versa. This change is not normally considered a bifurcation since there is no change in stability or creation of other states. Nevertheless, it is possible to identify surfaces in the parameter space that correspond to this transition. We call this transition the node focus (NF) transition.
Figure 12 reveals two surfaces: a lower surface with an internal pleat somewhat like a fortune cookie and an upper folded surface like the nose cone of an airplane. The PSS state occurs in the region in the far upper right corner of panels a and b. Here, the network dynamics is cooperative in that predominantly spiking neurons () interact via excitatory synapses (k>0), leading to an active network. In contrast, in the far lower left corner of these figures, predominantly resting neurons () interact cooperatively via inhibitory synapses (k<0), and the network exhibits the quiescent PSR state. The lower fortune cookie–like surface marks a NF boundary between these two regions, but the SN and AH bifurcations discussed above occur nearby as well. Interestingly, however, the upper nose cone surface encloses another region of PSR states. Within the nose cone, networks consist of predominantly resting but excitable neurons interacting via weak excitatory synapses. In this case, the resting states of most neurons are relatively far from their thresholds (), so that weak synaptic excitation is not sufficient to cause most neurons to fire. Thus, the network exhibits the PSR state.
It is important to note that the NF transition applies to both stable and unstable equilibria. In the one-dimensional bifurcation diagram of Figure 8b (with , ), the network transverses through both the lower and upper parts of the nose cone as k increases from 0 to 2.5. The intersection with the lower, fortune cookie–like surface occurs for a negative value of k, as indicated in Figure 8c. All these NF transitions involve stable equilibria.
In contrast, the PSR–PSS transition in Figure 11, visible in panel b, involves an unstable equilibrium. This transition corresponds to traversing the fortune cookie–like surface along the horizontal line shown in Figure 12c (at and k=−9; same as the horizontal lines in Figures 8 and 11). The NF transition occurs as the repeller (unstable PSR) that was created at the nearby saddle node bifurcation (at ) changes into an unstable focus (unstable PSS) at . Although these unstable PSR and PSS states are unobservable in the full network, the stabilization of the PSS state through the AH bifurcation at requires the preexistence of this unstable PSS state.
6. Summary and Discussion
Using the well-known theta neuron model, we constructed a heterogeneous network containing a mixture of at-rest but excitable neurons as well as spontaneously spiking neurons. These were globally coupled together through pulselike synaptic interactions whose shape depends on a sharpness parameter n. We applied a recently developed reduction technique to derive a low-dimensional dynamical equation that completely describes the asymptotic behavior of the network's mean field in the thermodynamic limit of large network size. By analyzing this reduced system, we identified not only all possible asymptotic states of the mean field, but also the bifurcations that occur as three network parameters are varied. We also showed that the predicted behavior derived in this way is exhibited by finite networks of 10, 000 neurons.
Of course, real neurons are not theta neurons. We caution that real type I neurons can be expected to be well approximated by theta neurons only near the onset of spiking. Future work will examine the extent to which our results carry over to networks of more realistic type I neurons. Nevertheless, the theta neuron model does capture important aspects of the generic behavior of type I neurons, including pyramidal neurons. These make up the majority of neurons in the mammalian brain (e.g., approximately 80% in the hippocampus and 70% in the cortex, across many species; Feldman, 1984; Chen & Dzakpasu, 2010). Also, real neuronal networks are clearly not globally coupled. But since our analysis focuses on the dynamics of the macroscopic mean field variable defined in equation 3.3, our results are best interpreted as providing an idealized mesoscopic description of neuronal dynamics (i.e., describing the behavior of a region of brain tissue that is small but nevertheless contains many neurons and many connections). Notable strengths of our network structure are that it includes heterogeneity in the internal dynamics of the individual neurons (i.e., both resting and spiking neurons) and that the coupling is via pulselike synapses of tunable sharpness that approximate real IPSPs without being delta functions. It is also important to note that our approach provides exact results for the asymptotic mean field behavior, although only in the limit of large system size. Nevertheless, we showed that the features and dynamics we derived are in fact exhibited by reasonably sized finite network instantiations.
We found that the asymptotic mean field of our network exhibits only three possible states: two corresponding to equilibrium solutions, the PSR and PSS states, and one limit cycle solution, the CPW state. The most obvious solution is perhaps the PSR state, in which the network consists of predominantly resting neurons that inhibit each other. In the most extreme case, the neurons form a single stationary cluster, and the mean field assumes a constant value. However, PSR and PSS states can also be realized such that all or a significant portion of the neuronal population spike regularly. Despite this spiking activity, the mean field for such partially synchronized states remains constant. These states are similar to asynchonous states that have been described by others (e.g., Abbott & van Vreeswijk, 1993; Hansel & Mato, 2001, 2003). For example, Abbott and van Vreeswijk (1993) defined the asynchronous state by the requirement that the total input into one neuron from all the others be constant, and they argued that the existence of these states justifies the use of firing rate models.
We have also described the subtle difference between the PSR and PSS states—that the former is a node and the latter is a focus. The consequences of this distinction can be observed in Figure 5, where it is shown that the mean field responses to perturbations are markedly different. When a PSR state is perturbed, the mean field relaxes to the equilibrium directly. In contrast, when a PSS state is perturbed, the mean field displays decaying oscillations. In this latter case, as the equilibrium microscopic configuration is approached, the neurons alternate between bouts of scattering and clumping, or desynchronization and resynchronization, in a manner such that each bout is less severe than the preceding one.
Similar microscopic dynamics underlie the CPW state, except that for the CPW state, the alternation between the de- and resynchronizing bouts persists indefinitely. Consequently, the asymptotic mean field approaches a limit cycle. This state is related to, but is more general than, the synchronous state described by Wang and Buzsáki (1996). The latter synchronous state occurs for homogeneous (or very weakly heterogeneous) networks when the phases of most neurons lock, as in the partially synchronous state shown in their Figure 5C. In this state, almost all neurons fire together. Thus, the order parameter of such a network exhibits a CPW state with a constant order parameter magnitude very close to one and a frequency of oscillation identical to that of an individual neuron. In contrast, the CPW states that we describe in this letter are more general in that typically, the neurons do not phase-lock. Instead, they form clusters of periodically varying coherence, as is quantified by the oscillating order parameter magnitude. The animation provided in the online supplement makes this point clear. It is important to note that the frequency of oscillation exhibited by the mean field while on a CPW limit cycle is an emergent property of the network as a whole. This frequency is therefore not related to the frequencies of the individual neurons in a simple manner. Indeed, in parameter space regions near the Bogdanov-Takens bifurcation, the CPW frequency can be very small. Such low-frequency rhythms have been reported in EEG (Avella Gonzalez et al., 2012), and slow population activities have been observed in hippocampal slice studies (Ho, Strüber, Bartos, Zhang, & Skinner, 2012).
These findings have implications for the interpretation of experimental measurements of network dynamics. For example, raster diagrams are often used to identify the presence or absence of synchrony. Our results suggest that such diagrams may not be sufficient to differentiate between the three macroscopic states that we have identified here.
The reduced equation, equation 3.13, also gives us the ability to fully analyze the relevant transitions among these macroscopic states using the degree of excitability, heterogeneity, and coupling strength (both excitatory and inhibitory) as bifurcation parameters. The results for two generic codimension 1 bifurcations, the saddle node and Andronov-Hopf bifurcations, were summarized visually as two-dimensional surfaces in Figures 7 and 9, respectively. We also examined the node focus transition similarly (see Figure 12). These results are exact only in the limit of large system size. In finite network realizations, fluctuations can build up when the network is near a bifurcation. Hence, the occurrence of these bifurcations can be expected to be blurred somewhat in the parameter space. Nevertheless, we found that the bifurcations we describe here can be reliably identified in networks of 10, 000 theta neurons. In this letter, we did not investigate smaller networks.
By examining these bifurcation surfaces, we found that the dynamical interplay between the internal neuronal dynamics, as parameterized by , and the interneuronal coupling, parameterized by k, appears to play an important role in determining the complexity of the possible macroscopic mean field dynamics of the network. In particular, when the network is preferentially cooperative, such that and k are of the same sign, the network tends to settle into one of the macroscopic equilibrium states. In contrast, when and k are of opposite sign, richer dynamics are seen, including the CPW state, bifurcations of the macroscopic mean field, and multistability.
Finally, we note that the results we reported were for two specific values of the sharpness parameter: n=2 and n=9. Further analysis of the effect of increasing this parameter shows no qualitative change in our results. Specifically, we found no changes to the three classes of macroscopic states and no significant qualitative changes to the bifurcation surfaces for values of n up to n=15. While we suspect that there may be important differences that arise as , we note that low values of the sharpness parameter are actually better approximations of real postsynaptic potentials.
Our results provide general predictions for the macroscopic dynamics of an idealized large network of type I neurons. In a related study (So et al., 2013), a similar but nonautonomous theta neuron network in which was made to oscillate in time was investigated. Varying this parameter such that the SNIC bifurcation is repeatedly traversed is a common method of modeling bursting neurons, specifically parabolic bursters. That letter builds on the results reported here and shows that more complicated behavior, including macroscopic quasiperiodicity, chaos, multistability, and final-state uncertainty, can occur in the nonautonomous case.
Future work will extend our results to include separate and different interacting populations of neurons, including type II neurons, in order to model the excitatory and inhibitory and layered structure of the brain's neuronal networks.
Most theoretical studies consider only these two stereotypical behaviors, but see Skinner (2013).
This method of analysis has been applied to other coupled networks of similar form (see, e.g., Pikovsky & Rosenblum, 2008, 2011; Abrams, Mirollo, Strogatz, & Wiley, 2008; Marvel & Strogatz, 2009; Martens, Barreto, Strogatz, Ott, So, & Antonsen, 2009; Abdulrehem & Ott, 2009; So & Barreto, 2011; Montbrió & Pazó, 2011; Alonso & Mindlin, 2011; Omel'chenko & Wulfrum, 2012; So, Luke, & Barreto, 2013).
Because the HC bifurcation is a global bifurcation, it cannot be specified by a set of simple algebraic conditions.