Such a cylindrical manifold, whose periodic boundary conditions embody the nature of the oscillatory dynamics, is shown in Fig. 4c. The convergence of the orbits shown in panels (b) and (c) onto the closed loop, on which they then remain, motivates the concept of an invariant set of the dynamics. Intuitively, this is simply a set of points (e.g. single point, closed loop, etc.) in which orbits remain once they enter. More formally if F represents the flow of a dynamical system then an invariant set A satisfies F(A) C A. The Van der Pol system in Fig. 4 has two invariant sets, one at the origin and the closed loop as shown. A variety of other orbits, from distinct initial conditions, are shown in panel (d). In each case, the orbits approach the limit cycle.

This simple observation motivates the crucial concept of an attractor, a bounded (i.e. finite) invariant set which is approached by the orbits from a "large set" of initial conditions. Traditionally, a large set implied an "open neighborhood" of the attractor. More recently the concept of an attractor has been generalized to mean any set with a non-zero probability measure (Milnor 1985) meaning that there is a (possibly very small but still non-zero) chance that an orbit from a randomly chosen initial condition will flow onto the attractor. On the other hand, there may be initial conditions arbitrarily close to the attractor that nonetheless flow elsewhere. This distinction is important in the setting of synchronization (Ashwin & Terry 2000) and we explore it further below.

We have hence seen fixed point and limit cycle attractors. A chaotic attractor has already been illustrated for the logistic equation in Fig. 1. In comparison to a limit cycle which endlessly repeats its prior states, a chaotic attractor never repeats a state although is nonetheless bounded and invariant. More formally, a chaotic attractor exhibits sensitive dependence to initial conditions - that is any two orbits, no matter how close initially - diverge at an exponential rate. This rate of divergence is captured by the largest characteristic exponent, which is positive for a chaotic attractor. In contrast,

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Neuronal Dynamics and Brain Connectivity (b)

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-fii D CL5 1 1.S 2 23

is -2 -1.4 -1 -0.6 0 0.5 t t.i 2 li z is -2 -1.4 -1 -0.6 0 0.5 t t.i 2 li

Fig. 4. Van der Pol oscillator for ^ = 2. (a) Time series. (b) Phase space portrait representation of a single orbit and its approach toward a limit cycle attractor. Arrows show representative vector field. (c) Representation of the same orbit on a cylindrical manifold spanned by the polar coordinates A(t) and 0(t). (d) A set of distinct initial conditions flowing toward the limit cycle attractor (black)

z z2

a z z two such points will stay forever close if on or near a limit cycle attractor (the largest characteristic exponent is zero). Two such points in the vicinity of a fixed point attractor will invariably get closer (the largest characteristic exponent is negative). A chaotic attractor is also topologically mixing - i.e. any given open set covering any region of the attractor will eventually overlap with any other region. The unceasing divergence of nearby orbits and the eventual mixing of regions combine to enable a chaotic attractor to be both unstable and bounded. We revisit chaos in the setting of specific neuronal models in Sect. 3.

An attractor's basin of attraction is the set of all initial conditions which have the attractor as their future state. In the present case, the basin of attraction for the loop is the entire plane, except for the origin. The inset of an attractor is that part of the basin of attraction with the strongest (principle)

direction of attraction. A repellor is an invariant set that is the past state of a large set of points, its basin of repulsion. The outset of a repellor is the subspace of this basin which diverges most quickly. In the present case, the origin is a repellor and its basin of repulsion is all the points within the loop.

If time was reversed in (20), the origin would become an attractor and points within the loop would be its basin of attractions. Points outside the limit cycle would diverge towards infinity. Hence the loop would be an example of a basin boundary or seperatrix. Basin boundaries can be repellers (as in the case here) or saddles which have an inset and an outset. A trivial example of a saddle is the origin in the system, dzi dz2

where, for a1 = 1, a2 = —1 and b1 = b2 =0 the z1-axis is the inset and the z2-axis is the outset (Fig. 5).

Occasionally, the outset from a repellor becomes the inset of an attractor. Such an entity, linking two fixed points, is called a heterocline. For a saddle, it is possible that the outset becomes the inset (due to curvature away from the saddle point). If so, it is called a homocline.

Two final concepts are required before we move onto more complex matters. An attractor possesses structural stability if it is insensitive to a small change in the nature of the vector field, corresponding to a small change in the evolution equation. "Insensitivity" here denotes that there exists a smooth mapping between the perturbed attractor and the original attractor. When such a mapping exists we say the two (original and perturbed) attractors are topologically conjugate. The saddle point in Fig. 5 is structurally stable following changes in either the parameters a (under a stretching and/or contraction) and/or the parameters b (under a translation). Similarly the Van der Pol attractor in Fig. 4 is structurally stable since small changes to any of the parameters results in another (topologically conjugate) limit cycle attractor.

The nullclines of a dynamical system are the curves in phase space, for which one derivative in the evolution equation is equal to zero, and hence correspond to the regimes in phase space with zero flow in a particular direction.

Fig. 5. Saddle point and orbits of (22)

Fig. 6. Nullclines of the Van der Pol system. Arrows show direction of the vector field across the nullclines. The fixed point lies at the intersection of the two mullclines (in blue and black). The trajectory is shown in red

The two nullclines to the Van der Pol equation (20) are depicted in Fig. 6. The blue line shows the nullcline for zero flow in the z\ direction and the black shows the curve for z2. The limit cycle trajectory satisfies these conditions as it crosses the respective curves - that is, the dz\/dt = 0 when the attractor (red curve) crosses the blue nullcline. By definition, any crossing of two nullclines corresponds to the existence and location of a fixed point since dzi/dt =dz2/dt = 0.

The nullclines form the "skeleton" of the phase space and, as we explore below, their intersections are vital to the existence and nature of most attrac-tors, not just fixed points.

2.2 Bifurcations and Complex Dynamics

The preceding discussion captures the nature of phase space dynamics and its relationship to the evolution equations for a given set of parameter values (i.e. when the vector field is kept constant). An intriguing and important field of study concerns what happens to the attractors and basin boundaries following a change to the system's parameters and hence to the vector field. From above it follows that if all the attractors are structurally stable, then the effect of such a change can be considered trivial since the dynamics will remain qualitatively similar (and typically also quantitatively similar). However, in the case when this is not so, sudden and dramatic changes in the dynamics, denoted bifurcations, occur. Examples abound in neuroscience, such as the generation of an action potential, the onset of bursting (Izhikevich 2005) and even the onset (Robinson et al. 2002, Lopes da Silva et al. 2003, Breakspear et al. 2006) and temporal progression (Rodriguez et al. 2006) of an epileptic seizure.

An important means of understanding the nature of a system's bifurcations is through the study of its bifurcation diagram. This is produced by smoothly varying one parameter over some range of interest whilst keeping all other parameters fixed. Hence the vector field is smoothly changed in one dimension of parameter space. At each parameter value, the system is integrated and, after passage of an initial transient - allowing for the system to evolve towards its attractor(s) - the asymptotic time series is captured. From this time series, the values of all local minima and maxima are stored. For a fixed point there will exist only one such value. For a simple (period-1) limit cycle there will exist two such points and for a period-2 oscillator, four such points - two maxima and two minima. For a chaotic oscillator, such points will be distributed densely ("almost everywhere") over one of more segments. The bifurcation diagram is the plot of these local maxima and minima against the respective parameter value. Figure 7 shows the bifurcation diagram of the logistic (4).

It is crucial to note that in most nonlinear systems, two or more attractors may co-exist for some parameter values, facilitating bistability or even mul-tistability. Each attractor will have basins, each separated by basin boundaries. In such cases, it is important that all such attractors are located when plotting a bifurcation diagram.

Bifurcations can be divided into local and global, as outlined below. Before doing so, it is important to introduce a second notion of stability. Structural stability concerns the robustness of invariant sets - attractors, repellors, saddles - to changes in the underlying vector field. In contrast asymptotic stability deals with the situation where the instantaneous state of the system is perturbed through addition of a small transient noise term (but the vector field is kept constant). An attractor is called asymptotically stable whenever the system returns towards the attractor following any such (small)

Fig. 7. Bifurcation diagram of the logistic equation (4). Top panel shows the local minima and maxima of the asymptotic time series against the parameter a. Lower three panels show representative time series (including the initial transient) with (1) fixed point, (2) limit cycle and (3) chaotic attractors. Note the "periodic windows" within the chaotic regime of the bifurcation diagram

Fig. 7. Bifurcation diagram of the logistic equation (4). Top panel shows the local minima and maxima of the asymptotic time series against the parameter a. Lower three panels show representative time series (including the initial transient) with (1) fixed point, (2) limit cycle and (3) chaotic attractors. Note the "periodic windows" within the chaotic regime of the bifurcation diagram noisy perturbation. A local bifurcation occurs whenever an attractor loses asymptotic stability whereas a global bifurcation corresponds to the loss of structural stability. These are also called subtle and catastrophic bifurcations (Abraham & Shaw 1988) because in the latter case the impact on the dynamics is typically more immediately discernable. We now explore such bifurcations in further detail.

Local Bifurcations

Local bifurcations concern the asymptotic stability of fixed point and other attractors. Consider the system governed by, f = AZ + B, (23)

where A is a matrix and B a vector. This is the matrix form of (22). Solutions in the case where B is zero are of the form

Hence the origin is a fixed point and the eigenvalues A of A determine the nature of the neighboring flow. Solutions in the case B = 0 are essentially the same after a suitable translation of the axes. The eigenvalues A = {Ai, A2} determine five possible types of fixed point systems (Fig. 8).

Figure 8(a) shows a typical flow when both eigenvalues are real and either both positive or both negative. Orbits diverge from (A1 > 0, A2 > 0) or converge to (A1 < 0, A2 < 0) the origin. In the former case, the fixed point is called a source and in the latter, a sink or node. We have already met the case (Fig. 8b) where the eigenvalues are real and opposite in sign (A1 > 0, A2 < 0) for the saddle point discussed above with regards to basin boundaries. When the eigenvalues are complex, they occur as complex conjugate pairs. The imaginary component endows the time series with an oscillatory component evident as spiraling orbits (Fig. 8c). When the real part of each eigenvalue is negative, these oscillations are damped and the fixed point is a spiral inset. Otherwise it is a spiral outset.

Fig. 8. Orbits for fixed points of the linear system (26). (a) Source or sink, (b) saddle, and (c) Spiral inset or outset

Whilst (26) is a simple linear system, the Hartman-Grobman theorem states that, for a very general class5 of nonlinear systems Fa, the flow within the neighborhood of a fixed point can be approximated by a suitable linear system with the form of (23). Hence these fixed points - and their stability -play an important role in many dynamical systems.

Note that the eigenvalues of A determine the divergence or convergence of nearby orbits. These are hence the "characteristic exponents" referred to in Sect. 2.1. In the setting of fixed points these are simply referred to as the eigenvalues of A. They are often called Floquet exponents in the vicinity of a limit cycle and Lyapunov exponents for a chaotic attractor. Following Eckmann and Ruelle (1984), we will simply refer to them as characteristic exponents, whatever the nature of the invariant set to which they refer.

Local bifurcations hence deal with the zero crossings of the characteristic exponents of attractors. The underlying set (typically) remains invariant, but loses its asymptotic stability. Just as a zero crossing can transform a fixed point from an attracting node into a saddle, the same also applies for both limit cycles and chaotic attractors. We now briefly discuss some of the canonical local bifurcations. In sect. 3 we will see how they relate to fundamental neuronal events such as firing and bursting.

Canonical Local Bifurcations

Pitchfork bifurcations occur when a single fixed point changes its (asymptotic) stability whilst also splitting off extra fixed points. In a supercritical pitchfork bifurcation a single stable fixed point attractor loses its stability as a parameter crosses its threshold and two new stable fixed points appear

(Fig. 9a). The evolution equation,

Fig. 9. Pitchfork bifurcation diagram (a) Supercriticial, and (b) subcritical. Solid line denotes fixed point attractor. Dashed lines denote fixed point repellors

5 As long as the derivative of Fa at the fixed point is not zero - i.e. the fixed point is hyperbolic.

yields this type of bifurcation at a = 0. Note that for a < 0, we have dz/dt < 0 when z > 0 and dz/dt > 0 when z < 0. Hence all initial conditions lead to the fixed point z = 0. Similar calculations show that when A crosses zero (a > 0) the origin becomes a source and fixed point attractors exist as ±i/a.

On the other hand the equation, yields a subcritical pitchfork bifurcation. In this case, the fixed point at-tractors at z = 0 also loses its stability as A crosses zero from below. However, two fixed point repellors exist at ±^f—a when a < 0 (Fig. 9b). Looking at the situation alternatively, one could say that the fixed point attractor at z = 0 loses its stability when two fixed point repellors collide with it at a = 0. However, in both cases, the fixed point remains an invariant of the system (i.e. dz/dt = 0) for all a.

In a transcritical bifurcation, there are two equilibrium points which collide and exchange their stability at the bifurcation point. For example, the evolution equation, has two equilibrium points, the origin x = 0 and x = a. When a < 0, the origin is an attractor but becomes a repellor as a crosses zero (Fig 10).

A Hopf bifurcation (Fig. 11) is much like a pitchfork bifurcation with the exception that it involves a limit cycle attractor. Hopf bifurcations play an important role in neuronal models as they describe the onset of both sub-threshold membrane oscillations and cell firing. Consider the equation, where a is the bifurcation parameter and both z and b are complex numbers. When the real part of b is negative then the system exhibits a supercritical Hopf bifurcation (Fig. 11a,c-f). For a < 0 there exists a single stable fixed point attractor (a spiral inset). When a > 0 this fixed point is an unstable spiral outset and there also exists a stable limit cycle.

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