Burst Synchronization

• Dr. Jonathan E. Rubin, University of Pittsburgh

Defining burst synchronization

General definition

Neurons engage in various forms of activity, characterized by the generation of particular patterns of action potentials. One such activity pattern is a burst, which consists of a group of at least two action potentials that occur relatively close together in time (the active phase ), separated from all other action potentials by sufficiently large time intervals (silent phases ). Various classifications of bursting patterns for individual neurons have been established, depending on the minimal models that produce them (e.g. Rinzel, 1987) or on the bifurcation events leading to the onset and offset of the active phase (Izhikevich, 2000). Some of these, along with a variety of biological examples and further issues, are reviewed elsewhere in this encyclopedia (see Bursting).

A variety of different definitions of synchronization appear in the neuroscience literature. In a weak sense, neurons can be considered to exhibit synchrony if there is some consistent temporal relationship between some aspects of their respective activity patterns. Following this notion, synchronization of neurons refers to the establishment of some such relationship between them. Synchronization may, for example, be achieved through direct coupling or through common inputs. Some authors impose stronger definitions of synchrony, requiring the maintenance of a specific phase relationship between events (e.g. neuronal action potentials) or even a precise temporal coincidence of events. Commonly mentioned phase relationships include in-phase, which refers to events that happen together, and anti-phase, which is used to describe events that alternate in time.

Based on these ideas, burst synchronization naturally refers to the introduction of a temporal relationship between the bursts produced by two or more neurons. This definition includes some ambiguity, because the active phase of each burst consists of multiple spikes. Hence, the phrase burst synchronization is typically used to refer to a temporal relationship between active phase onset or offset times across neurons, while spike synchronization characterizes a temporal relationship between the spikes fired by different bursting neurons within their respective active phases. For instance, a pair of cells can exhibit in-phase burst synchronization with anti-phase spiking, meaning that they enter and exit the active phase together, yet during their shared active phase, they take turns spiking.

This form of synchronization can be observed, for example, when either excitatory synaptic coupling or diffusive coupling is introduced between a pair of model respiratory neurons (de Vries and Sherman, 2005; Butera et al., 2005); see Figure 1

Figure 1: Nearly in-phase (top) and anti-phase (bottom) spike synchronization within synchronized bursts.

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Experimental relevance

Experimentally, burst synchronization, in one sense or another, has been considered within cell cultures, where the interaction of spontaneous bursts, stimulation-induced bursts, and propagation of activity can be conveniently studied (e.g. Maeda et al., 1995). Synchronization of bursts may also be particularly relevant in responses to novel stimuli (Sherman, 2001), in thalamocortical interactions in the context of sleep rhythms (Steriade et al., 1997), and in pathological conditions. For example, an increase in burstiness and synchronization has been observed in various areas of the basal ganglia in Parkinson's disease, and it has been hypothesized that these developments contribute to the motor complications, particularly resting tremor, that are characteristic of this condition (Bergman et al., 1998; Bevan et al., 2002). Bursting also has been observed in non-neuronal cell types. In particular, significant theoretical work has focused on the analysis of bursting in insulin-secreting pancreatic \(\beta\)-cells.

Elements contributing to burst synchronization

The activity pattern that develops when coupling is introduced between two or more neurons depends on their intrinsic dynamics as well as the nature of the coupling. There are a large variety of possible burst mechanisms that can arise intrinsically in single cell models. These mechanisms can be classified by extracting the fastest evolving equations within a model (the fast subsystem) and considering the bifurcation mechanisms in the dynamics of these equations that yield the onset and offset of the active phase (Rinzel, 1987; Izhikevich, 2000a). For example, the most commonly studied form of bursting is known as square-wave bursting. While the name square-wave bursting refers to the approximately constant spike amplitude seen during the active phase (see Figure 2 below), this solution is also known as fold/homoclinic bursting, since the onset of the active phase occurs at a fold bifurcation for the fast subsystem and the offset of the active phase occurs at a homoclinic bifurcation for the fast subsystem.

Coupling between cells, in general terms, can be diffusive, via gap junctions or electrical synapses, or synaptic, through chemical synapses. Synaptic coupling can be excitatory or inhibitory, fast or slow, and depressing, facilitating, or neither. In every one of these cases, parameters can be tuned such that some form of burst synchronization results when coupling is introduced between two identical, intrinsically bursting model cells. Moreover, under some conditions, model cells that do not burst in the absence of coupling can be induced by coupling to burst in a synchronized way, and heterogeneity or noise can promote such emergent bursting. In the next part of this article, several examples are presented to illustrate these theoretical results.

Burst synchronization in particular settings

A general theory or classification scheme on burst synchronization has not been developed. The examples in the following sections represent particular results that have been observed. It is important to note that most of these results depend on model details or even parameter choices within models, and different outcomes may occur when these specifics are varied.

Diffusive coupling

A neuron in a diffusively coupled network may, for example, be modeled as \[ \begin{array}{rcl} C\dot{V_i} & = & f(V_i,h_i) + \sum_{j \neq i}^N g_{ji} (V_j-V_i), \\ \dot{h_i} & = & g(V_i,h_i) \end{array} \] where `V_i denotes voltage, the terms being summed represent the coupling from other cells in the network, indexed by j and scaled with constants g_ji, and h_i is a vector encompassing a collection of ion channel activation and inactivation variables. In a pair of identical intrinsically square-wave bursting cells, the introduction of such coupling, with g₁₂=g₂₁=g>0, leads to the co-existence of two solutions exhibiting in-phase burst synchronization, one with in-phase spikes and one with anti-phase spikes. In some cases, for small g, the solution with anti-phase spikes is stable and the other is not. As g is increased, however, stability switches to the in-phase branch (Sherman and Rinzel, 1992). In other cases, the anti-phase spiking solution is never stable (de Vries and Sherman, 1998).

It is also possible that diffusive coupling could appear in an equation other than the voltage equation, such as an equation for the intracellular concentration of calcium or some other ion, representing a diffusive exchange of that ion between cells. Weak diffusive coupling of calcium, for example, has been shown to enhance burst synchronization in a pancreatic \(\beta\)-cell model, but stronger coupling was found to yield a death of oscillations through a pitchfork bifurcation. This mechanism arises in models in which calcium drives a negative feedback, and it carries over even if diffusive ionic coupling and diffusive voltage coupling are both present, as long as the latter is not too strong relative to the former (Tsaneva-Atanasova et al., 2006).

Relaxation oscillations

Models exhibiting bursting often can be decomposed into subsets of variables that evolve on highly disparate timescales. For example, given \[\tag{1} \begin{array}{rcl} \dot{V} & = & f(V,h), \\ \dot{h} & = & \epsilon g(V,h), \end{array} \]

if \( \epsilon>0 \) is sufficiently small, then V is the fast variable and h is the slow variable. The fast subsystem corresponding to system (1) is the equation \( \dot{V} = f(V,h) \) with h constant. The slow subsystem for (1) is \( h'=g(V(h),h)\ ,\) where \( f(V(h),h)=0 \) and differentiation is with respect to rescaled time.

To simplify analysis of burst synchronization in such systems, it is sometimes useful to eliminate terms responsible for fast spiking, as long

as this can be done in a way that preserves oscillations between silent and active phases at burst frequency ( Figure 2)

Figure 2: Bursts with (top) and without (bottom) spikes.

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Oscillatory solutions of the resulting reduced model that make such transitions are called relaxation oscillations. Cells in model neuronal networks consisting of relaxation oscillators can be coupled diffusively or synaptically. If the variable \( V \) in system (1) denotes voltage, then synaptic coupling may be modeled through the addition to the \( V \) equation of either

• an explicitly time-dependent term depending on the

spike times of presynaptic cells, such as \( \sum_j \alpha(t-t_j) \) for spike times t_j, where \( \alpha(t) \) is some function that is zero for t<0,

• a term depending directly on presynaptic voltage, or

• a term depending on an auxiliary variable that obeys its own

differential equation that is driven by the presynaptic voltage.

In each of the last two cases, we can write the synaptic term in (1) as \(c(V_{pre},V)\ ,\) since in the third case the auxiliary variable depends implicitly on \(V_{pre}\ ;\) however, unlike the second case, the third case introduces an additional timescale, associated with the auxiliary variable, into the system.

A key mechanism affecting synchronization when synaptic coupling turns on quickly is fast threshold modulation (FTM) (Somers and Kopell, 1993). In relaxation oscillations, the transition of a cell between its silent and active phases occurs when the cell reaches a saddle-node bifurcation (or knee) in a manifold of equilibrium points of its fast subsystem, under the flow of its slow subsystem. Suppose that when synaptic coupling is introduced, system (1) becomes \[ \begin{array}{rcl} \dot{V} & = & f(V,h)+c(V_{pre},V), \\ \dot{h} & = & \epsilon g(V,h), \end{array} \] where \( c(V_{pre},V) \) represents a coupling function of the second or third type described above. If the value of \( c(V_{pre},V) \) changes abruptly, then the manifold of equilibrium points where \( \dot{V}=0 \) may also jump abruptly in phase space. In particular, the knee that forms the threshold for transitions between phases moves rapidly, hence FTM occurs. In FTM, for example, a cell that is below a knee for \( c(V_{pre_1},V)=c_1 \) may lie above the corresponding knee for \( c(V_{pre_2},V)=c_2, \) such that if the coupling term quickly changes from c₁ to c₂, the cell quickly

jumps between phases ( Figure 3)

Figure 3: The cell is below the knee for coupling c₁ but above the knee for coupling c₂

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FTM can lead to very rapid synchronization of relaxation oscillators, although it does not guarantee that the synchronization will be in-phase. Indeed, it is interesting to note that in-phase and out-of-phase synchronization of relaxation oscillators can both be induced by both excitatory and inhibitory synaptic coupling, depending on model details, including synaptic timescales (see Rubin and Terman, 2002, for a partial review).

Synaptic coupling

A classical example of phase-locked bursting, specifically anti-phase burst synchronization, is the half-center oscillation (Brown, 1914). In this rhythm, one burster is in the active phase while the other is in the silent phase, and at some point the bursters switch roles. A half-center oscillation may be achieved by coupling two bursters with inhibitory synapses, which ensures that their active phases do not overlap. Alternatively, a half-center oscillation may emerge from coupling two continuously spiking cells with synaptic inhibition, if either 1) each cell undergoes some form of adaptation while it is spiking, such as spike slowing through a gradually augmenting outward current, that allows the suppressed cell to become active, or 2) each cell includes some feature, such as the hyperpolarization-induced deinactivation of an inward current, that allows it to escape from inhibition and become active after a period of suppression.

When a pair of identical square-wave bursters are coupled with fast excitatory synapses, burst synchronization has been found to occur. Analogously to the case of diffusive coupling, the stable solution in this setting may feature anti-phase spikes. The effect of increasing the coupling strength is apparently model-dependent; the anti-phase spiking state may remain stable until a transition to tonic spiking occurs (Best et al., 2005) or the in-phase spiking state may take over (de Vries and Sherman, 2005). In several models featuring bursts composed of small numbers of spikes as well as nonzero synaptic delays, stable in-phase spike synchronization within bursts has also been observed. It was proposed that the key to this result is arrival of an input in between spikes, when the trajectory of the postsynaptic cell is close to its voltage nullcline (or nullsurface) and hence a large phase advance can be induced (Takekawa et al., 2007). The precise role of synaptic delays in this result remains to be established.

Applying a sufficiently strong excitatory synaptic input with constant conductance to an uncoupled square-wave burster can switch its activity to tonic spiking. Given this, a seemingly paradoxical result is that after such an input is applied to two identical square-wave bursters and induces tonic spiking, the introduction of excitatory synaptic coupling between the neurons can in fact cause the pair to switch back to bursting, in a synchronized manner (Butera et al., 1999). This observation can be explained in terms of a fast-slow decomposition and bifurcation analysis (Best et al., 2005). Such promotion of bursting by excitatory synaptic coupling may be relevant for respiratory rhythms (Butera et al., 2005).

In networks of bursters, existence and stability results for particular synchronized solutions can be obtained using consistency conditions based on phase resetting curves (PRC). A basic PRC is a function \( \Delta(\phi) \ ,\) such that if a perturbation to an oscillator occurs when that oscillator is at phase \( \phi \) of its oscillation, then its phase is shifted by \( \Delta(\phi)\ .\) In a network of coupled bursters, however, an input to one burster may shift the duration of its active phase, and hence the duration of the input to all cells postsynaptic to that one, which complicates the analysis. Nonetheless, this analysis can be carried out in small networks as long as the effects of each input to a cell are confined to the two burst cycles following its occurrence (Canavier, 2005).

Larger networks and arbitrary coupling dynamics

In larger coupled networks of bursters, clustered solutions may exist, in which cells within each cluster are in-phase synchronized, while different clusters take turns entering the active phase. If the cells within each cluster are in fact synchronized with zero phase differences, then the number of cells in each cluster is relevant for solution existence only inasmuch as this feature affects the strength of coupling between clusters. To analyze the stability of a particular clustered solution, it is necessary to consider the robustness to perturbations of both the synchronization of the cells within each cluster and the phase differences between clusters. Clustered bursting oscillations, with in-phase synchrony within each cluster, have been proposed as a binding mechanism. According to this idea, neurons that encode a particular stimulus feature synchronize in the same cluster. For example, if a red vertical bar were observed, then cells responding to the color red and cells responding to vertical bars would engage in in-phase synchronized bursting oscillations together (Terman and Wang, 1995).

Certain results about synchronization can be derived based only on the topology of the coupling architecture in networks composed of identical neurons or of multiple classes of identical neurons (e.g. Belykh et al., 2005; Golubitsky et al., 2005). Typically, these findings concern perfectly in-phase synchronization or clustered oscillations with exact in-phase synchrony within clusters and some form of symmetry of phase relations between clusters. For example, a synchronous solution will exist if a network is balanced, in the sense that in the synchronous state, the total input strength to each cell in the network from other cells in the network is the same. Since these results do not depend on the particular dynamics of the identical cells, as long as they oscillate, they apply to bursting oscillations in particular, but not exclusively. In contrast, one result that is bursting-dependent but is still independent of the details of synaptic coupling has been obtained for a pair of elliptic bursters with sufficiently similar spike frequencies. Analysis in this setting, based on a normal form for a Bautin bifurcation, shows that weak coupling can be sufficient to lead to rapid burst synchronization, regardless of the form of the coupling, through either FTM or elimination of a delayed bifurcation effect. In the normal form analysis, a two-timescale canonical model is derived in which the coupling terms are linear combinations of the fast variables. In this setting, certain additional details, such as the phase relationships between spikes within the synchronized bursts, do depend on the form of coupling, which determines the signs of the coupling coefficients in the canonical model(Izhikevich, 2000b).

Alternatively, connection architectures lacking symmetry may produce more complex dynamics featuring some degree of burst synchronization. For example, switching a network of Morris-Lecar cells, driven by a pacemaking core, from a local to a small-world to a random connectivity was observed to induce a switch from propagating waves to somewhat synchronized bursting to synchronized spiking in response to each pacemaker oscillation (Shao et al., 2006).

Heterogeneous networks

Heterogeneous networks can be formed by coupling together cells with different forms of intrinsic dynamics. With diffusive or synaptic coupling, under the right conditions, a wide variety of heterogeneous networks can each engage in synchronized bursting. Even a network composed of an intrinsically quiescent cell and an intrinsically tonically spiking cell can exhibit synchronized bursting if either diffusive or excitatory synaptic coupling is introduced between the cells, although the dynamic mechanisms differ in the two cases. In the synaptic case, the synchrony may fail to be precise, in the sense that the cells may enter and exit the active phase at different times, but synchrony is achieved in that a consistent temporal relationship between the cells' activity patterns develops. When the introduction of coupling induces bursting in a network of cells that do not burst when uncoupled, this phenomenon is referred to as emergent bursting (de Vries and Sherman, 2005). While the presence of noise was found to enhance emergent bursting in a coupled network (de Vries and Sherman, 2000), the mechanism underlying this finding is effectively that noise introduces another source of heterogeneity into the system (Pedersen, 2005).

Burst synchronization can also be considered in networks featuring combinations of excitatory and inhibitory synaptic coupling (E-I networks). For example, suppose a synchronized group E₁ of excitatory bursters becomes active and induces activity in a collection I₁ of inhibitory cells to which they send synaptic inputs. If the pattern of inhibitory connections is off-register with the excitatory coupling architecture, then group I₁ will inhibit a set E₂ of excitatory cells that is disjoint from E₁. Eventually, E₂, along with an associated group I₂ of inhibitory cells, may replace E₁ and I₁ in the active phase, and a clustered bursting solution results. E-I networks can engage in a variety of different bursting activity patterns, featuring different degrees of synchronization, depending on the relative coupling strengths, coupling architecture, and intrinsic dynamics represented in the network. As noted earlier, changes in these patterns may be relevant to changes in sleep states, in the setting of thalamocortical networks, or to changes associated with parkinsonian dopamine depletion, in the indirect pathway of the basal ganglia.

In networks composed of segregated columns of intrinsic bursters together with columns of regular or tonically spiking cells, it has been observed that the introduction of a small set of long range connections enhances burst synchronization across the network (French and Gruenstein, 2006). To a large extent, however, bursting in heterogeneous networks of cells with complex coupling architectures remains to be analyzed.

Burst synchronization via common inputs

Results have begun to emerge on synchronization of uncoupled neurons through common or correlated inputs. This work has not focused on bursting neurons. In other fields, such as the study of laser dynamics, this topic has received some attention, including experimental studies. For example, a common noise input has been shown to induce some degree of burst synchronization in similar, but non-identical, bursting lasers, and this effect was enhanced by sinusoidal modulation of the input signal (DeShazer et al., 2004).

Maps

A variety of maps have been used as computationally efficient representations of neuronal dynamics. These maps may be designed, or derived, to allow the possibility of bursting solutions. For example, in square-wave bursting in a differential equation model (e.g. Chay and Rinzel, 1985), a slow variable may increase during the silent phase and then show a net decrease during each oscillation in the active phase. In an analogous burst solution in a map, the slow variable would gradually decrease over a sequence of iterations, each corresponding to one spike in the active phase. When the slow variable became sufficiently small, its value would jump up, providing a

condensed representation of reinjection into the active phase after passage through the silent phase (see Figure 4; see also Medvedev, 2005)

Figure 4: A schematic of a 1-d bursting map (black) with a superimposed bursting trajectory (blue).

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As with differential equation models, coupling may be introduced between map-based bursters, and burst synchronization, featuring various phase relationships, may occur (de Vries, 2005). It has not yet been established, however, what the precise transformations are from forms of coupling studied in differential equation models to coupling terms appearing in maps.

Open Questions

The consideration of burst synchronization in networks of more than two cells, particularly featuring connection architectures other than nearest-neighbor or all-to-all, remains as a largely open area of research. The many open challenges in the study of burst synchronization include

• the analysis of burst synchronization in networks of synaptically coupled non-square-wave bursters,
• the rigorous establishment of conditions for the stability of synchronized bursting solutions with anti-phase or in-phase spikes in networks of two or more cells with diffusive or excitatory synaptic coupling,
• the exploration of how coupling delays interact with other network features to affect burst synchronization
• the systematic analysis of burst synchronization in heterogeneous networks of more than two cells, including the interaction of intrinsic dynamics and coupling architecture,
• the rigorous analysis of how burst characteristics, such as period and duty cycle, depend on coupling strength and other parameters in coupled networks with synchronized bursting solutions,
• the analysis of the effects of noise on coupled networks of bursters and on emergent bursting,
• the exploration of burst synchronization in uncoupled or weakly coupled cells sharing a common external input or correlated external inputs, and
• the interaction of multiple mechanisms, each of which is sufficient to induce synchronized bursting on its own, within a coupled neuronal network.

References

• Belykh I., de Lange E., and Hasler M. (2005) Synchronization of bursting neurons: What matters in the network topology. Phys. Rev. Lett. 94: 188101.
• Bergman H., Feingold A., Nini A., Raz A., Slovin H., Abeles M., and Vaadia E. (1998) Physiological aspects of information processing in the basal ganglia of normal and parkinsonian primates. Trends in Neurosci. 21:32-38.
• Best J., Borisyuk A., Rubin J., Terman D., and Wechselberger M. (2005) The dynamic range of bursting in a model respiratory pacemaker network. SIAM J. Appl. Dyn. Sys.
• Bevan M., Magill P., Terman D., Bolam J., and Wilson C. (2002) Move to the rhythm: oscillations in the subthalamic nucleus-external globus pallidus network. Trends Neurosci. 25:525-31.
• Brown T.B. (1914) On the nature of the fundamental activity of the nervous centers; together with an analysis of the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the nervous system. J. of Physiology. 48: 18-46.
• Butera R., Rinzel J., and Smith J. (1999) Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Populations of coupled pacemaker neurons. J. Neurophysiol. 81:398-415.
• Butera R., Rubin J., Terman D., and Smith J. (2005) Oscillatory bursting mechanisms in respiratory pacemaker neurons and networks. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 303-346.
• Canavier C. (2005) Analysis of circuits containing bursting neurons using phase resetting curves. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 175-200.
• Chay T. and Rinzel J. (1985) Bursting, beating, and chaos in an excitable membrane model. Biophys. J. 47:357-366.
• DeShazer D., Tighe B., Kurths J., and Roy R. (2004) Experimental observation of noise-induced synchronization of bursting dynamical systems. IEEE Jour. Sel. Top. Quant. Elect. 10: 906-910.
• de Vries G. (2005) Modulatory effects of coupling on bursting maps. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 223-242.
• de Vries G., Sherman A., and Zhu H.-R. (1998) Diffusively coupled bursters: effects of cell heterogeneity. Bull. Math. Biol. 60: 1167-1200.
• de Vries G. and Sherman A. (2005) Beyond synchronization: modulatory and emergent effects of coupling in square-wave bursting. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 243-272.
• French D.A. and Gruenstein E.I. (2006) An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons. J. Comp. Neurosci. 21:227-241.
• Golubitsky M., Josic K. and Shiau L. (2005) Bursting in coupled cell systems. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 201-222.
• Izhikevich E.M. (2000a) Neural excitability, spiking, and bursting. Int. J. Bif. and Chaos, 10:1171-1266.
• Izhikevich E.M. (2000b) Synchronization of elliptic bursters. SIAM Review, 43:315-344.
• Maeda E., Robinson H., and Kawana A. (1995) The mechanisms of generation and propagation of synchronized bursting in developing networks of cortical neurons. J. Neurosci. 15:6834-6845.
• Medvedev G. (2005) Reduction of a model of an excitable cell to a one-dimensional map. Physica D, 202:37-59.
• Pedersen M. (2005) A comment on enhanced bursting in pancreatic beta-cells. J. Theor. Biol. 235: 1-3.
• Rinzel J. (1987) A formal classification of bursting mechanisms in excitable systems. In: A. Gleason, ed. Proceedings of the International Congress of Mathematicians, American Mathematical Society, Providence, RI, 1578-1594.
• Rubin J. and Terman D. (2002) Geometric singular perturbation analysis of neuronal dynamics. In: B. Fiedler, ed. Handbook of Dynamical Systems, vol. 2: Towards Applications. Elsevier, Amsterdam, 93-146.
• Shao J., Tsao T.-H., and Butera R. (2006) Bursting without slow kinetics: A role for small world? Neural Comput. 18: 2029-2035.
• Sherman A. and Rinzel J. (1992) Rhythmogenic effects of weak electrotonic coupling in neuronal models. Proc Natl Acad Sci U S A. 89:2471-4.
• Sherman S.M. (2001) Tonic and burst firing: dual modes of thalamocortical relay. Trends in Neurosci., 24:122-126.
• Somers D. and Kopell N. (1993). Rapid synchrony through fast threshold modulation. Biol. Cybern. 68: 393-407.
• Steriade M., Jones E. and McCormick D. (editors, 1997) Thalamus, Volume II, Elsevier, Amsterdam.
• Takekawa T., Aoyagi T., and Fukai T. (2007). Synchronous and asynchronous bursting states: Role of intrinsic neural dynamics. J. Comp. Neurosci. 23:189-200.
• Terman D. and Wang D. L. (1995). Global competition and local cooperation in a network of neural oscillators. Physica D 81: 148-176.
• Tsaneva-Atanasova K., Zimliki C., Bertram R., and Sherman A. (2006) Diffusion of calcium and metabolites in pancreatic islets: Killing oscillations with a pitchfork. Biophys. J. 90:3434-3446.

Internal references

• Peter Redgrave (2007) Basal ganglia. Scholarpedia, 2(6):1825.
• John Guckenheimer and Yuri A. Kuznetsov (2007) Bautin bifurcation. Scholarpedia, 2(5):1853.
• John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
• Wolf Singer (2007) Binding by synchrony. Scholarpedia, 2(12):1657.
• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• Peter Jonas and Gyorgy Buzsaki (2007) Neural inhibition. Scholarpedia, 2(9):3286.
• James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Carmen C. Canavier (2006) Phase response curve. Scholarpedia, 1(12):1332.
• Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
• Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.
• Leonid L. Rubchinsky, Alexey S. Kuznetsov, Vicki L. Wheelock MD, Karen A. Sigvardt (2007) Tremor. Scholarpedia, 2(10):1379.

See also

Bursting, Synchronization, Fast Threshold Modulation, LEGION, Neural Synchrony Measures

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Anonymous

Accepted on: 2007-10-11 02:23:46 GMT