Typically $ is taken to be a constant, i.e., x(t) = x0, —t < t < 0 , which is reasonable for most experimental systems. It should be noted that only solutions which are asymptotically stable can be accurately approximated using numerical integration.
There are two main programs available for the numerical integration of delay differential equations. The widely-used (and free) package XPPAUT (Ermentrout, 2005) can perform numerical integration using a variety of fixed step numerical methods, including Runge-Kutta. It has a good graphical user interface for visualizing the results. Perhaps the most useful aspect of this program is the ease with which parameters and initial conditions can be changed. The recent book of Ermentrout (2002) gives a overview of the package including many examples. Information on how to download the package as well as documentation and tutorials are available at www.math.pitt.edu/~bard/xpp/xpp.html. Within MATLAB there is the function DDE23 (Shampine and Thompson, 2001) which is a variable step size numerical integration routine for delay differential equations. A tutorial is on this routine available at www.mathworks.com/dde_tutorial. Results maybe visualized using the extensive graphing tools of MATLAB.
Numerical bifurcation analysis consists of two parts, the approximation of a solution and the calculation of the stability of this solution. The approximation of a solution in a numerical bifurcation package is not done using numerical integration, but rather using numerical continuation. Numerical continuation uses a given solution for a particular parameter value to find a solution for a different (but close) parameter value. This is only easily implemented for equilibrium and periodic solutions. Both stable and unstable solutions can be found. Once an equilibrium solution is found to a desired accuracy, approximations for a finite set of the eigenvalues with the largest real part can be determined, which will determine the stability of the equilibrium point. The stability of periodic orbits can be numerically determined in a similar way. Numerical bifurcation packages generally track the stability of equilibrium points and periodic orbits, indicating where bifurcations occur.
There is one package available that does numerical bifurcation analysis for delay differential equations, DDE-BIFTOOL (Engelborghs et al., 2001). This package runs on MATLAB. An overview of the numerical methods used in this package and some examples applications can be found in the paper of Engelborghs et al. (2002). The user manual and information on how to download the package are available at
3 Effects of delay
In this section I will outline some of the effects of delay that have been documented in the literature.
Time delays are commonly associated with type II oscillations, i.e. oscillations created by a Hopf bifurcation (Breakspear and Jirsa, 2006), for the following reason. There are many examples of systems that have a stable equilibrium point if the time delay is zero (or sufficiently small), but have oscillatory behaviour if the delay is large enough. In these systems, the oscillation is created via a Hopf bifurcation at a critical value of the delay. This is sometimes referred to as a delay-induced oscillation. One of the simplest examples of this is the following model for recurrent inhibition due to Plant (1981):
v(t) = v(t) — 1 v3(t) — w(t) + c(v(t — t ) — vo) W(t) = p(v(t) + a — bw(t)) .
This is a Fitzhugh-Nagumo model neuron with a delayed term which represents recurrent feedback. Plant considered parameters such that the system with no feedback has a stable equilibrium point and showed that this stability is maintained for the system with feedback and sufficiently small delay. He then showed that when c < 0 (i.e. the recurrent feedback is inhibitory), there is a Hopf bifurcation at a critical value of the delay, leading to oscillations.
One of the most publicized (Strogatz, 1998) effects of time delays is the fact that the presence of time delays in the coupling between oscillators can destroy the oscillations. This phenomenon, usually called oscillator death or amplitude death was first noted by Ramana Reddy et al. (1998), in their analysis of a simple model of type II oscillators with gap junctional coupling. Subsequently Ramana Reddy et al. (2000) observed this phenomenon experimentally in a system of two intrinsically oscillating circuits with the same type of coupling. There are many papers related to delay induced oscillator death in the coupled oscillator literature, which I will not attempt to review here. Instead I will focus the discussion on results relevant to neural models.
The work of Ramana Reddy et al. (1998, 1999) shows that when two or more intrinsically oscillating elements are connected with gap junctional coupling of sufficient strength with a sufficiently large delay then the oscillations may be destroyed. Their work focused on systems where the elements were identical except for the frequency of the intrinsic oscillations and the coupling was all-to-all and symmetric (all the coupling coefficients were the same). Their model oscillator was just the normal form for the Hopf bifurcation. This behaviour has also been seen for a delayed, linearly coupled (i.e. (3) with no xj(t) term) pair of van der Pol oscillators (Wirkus and Rand, 2002), and for a pair of Fitzhugh-Nagumo oscillators with delayed gap junctional coupling (Campbell and Smith, 2007). To my knowledge this has yet to be observed for other biophysical models of neural oscillators, however, it may be expected to occur for most type II oscillators. Atay (2003b) obtained results for a network of weakly nonlinear oscillators with a symmetric connection matrix and gap junctional coupling. He showed that if the intrinsic frequency of the oscillators is sufficiently similar then oscillator death can occur.
Several studies have shown that the type of oscillator death described above does not occur for type II oscillators with sigmoidal coupling (Buric and Todorovic, 2003; Campbell et al., 2004; Shayer and Campbell, 2000). However, a different type of oscillator death can occur (Buric and Todorovic, 2003; Buric et al., 2005; Campbell et al., 2004; Shayer and Campbell, 2000): for elements which are intrinsically excitable (i.e. not oscillating when decoupled), oscillations induced by instantaneous coupling may be lost if a time delay is introduced.
The work of Buric et al. (2005) has shown that for the type I oscillator of Terman and Wang (1995), there is no oscillator death of this latter type with either gap junctional or sigmoidal coupling. Their work also suggests that delay induced oscillator death of the first type is not possible.
The study of type II oscillator death in coupled neural systems combines various techniques of Sect. 2. Oscillator death can occur when increasing the time delay causes the stabilization of an equilibrium point. Values of the delay where this occurs will correspond to places where the characteristic (19) has an eigenvalue with zero real part and dR^X) < 0. To have oscillator death, however, one must also show that the periodic orbit is eliminated. This means that at the value of t where the equilibrium point stabilizes, there is a "reverse" Hopf bifurcation destroying the stable limit cycle. This may be checked via numerical simulations or numerical continuations (see subsection 2.5), or by showing, as outlined in subsection 2.2, that the Hopf bifurcation is subcritical. Buric et al. (2005) and Buric and Todorovic (2003, 2005) have shown that for excitable Fitzhugh-Nagumo neurons, the restabilization of the equilibrium point is not always accompanied by oscillator death. In the case that the Hopf bifurcation is subcritical, the stable oscillator may persist with the stable equilibrium point giving a region of bistability. In their model, for larger values of t the periodic orbit is eliminated in a saddle-node bifurcation of limit cycles, leading to oscillator death.
The results of Buric et al. on type I oscillator death are primarily based on numerical simulations. To my knowledge there has been virtually no mathematical study of this situation. Recall that type I oscillators are those where the oscillation is created by an infinite period bifurcation (Breakspear and Jirsa, 2006). If such a bifurcation takes place in the coupled system with no time delay, introducing a time delay will not change the presence of the saddle-node bifurcation, however, it may affect whether this bifurcation occurs on an invariant circle. Continuity arguments would suggest that for sufficiently small delay, the saddle-node bifurcation will still occur on the invariant circle, leading to the creation of a periodic orbit at exactly the same bifurcation point as for the undelayed system. What happens for large delay remains to be investigated.
A significant observation about ANNs of the form (14), is that many intersections between different Hopf bifurcation curves and between Hopf bifurcation curves and pitchfork bifurcation curves can occur (Belair et al., 1996; Shayer and Campbell, 2000; Yuan and Campbell, 2004). Figure 1 shows that this occurs in our coupled Fitzhugh-Nagumo model as well. These intersection points are called codimension two bifurcation points. Such points can lead to more complicated dynamics including: the existence of solutions with multiple frequencies (quasiperiodicity), the coexistence of more than one stable solution (multistability) or the switching of the system from one type of solution to another as a parameter is varied (Guckenheimer and Holmes, 1983, Chap. 7), (Kuznetsov, 1995, Chap. 8). In ordinary differential equations, such points are quite rare. In delay differential equations, however, such points are more common as the time delay forces there to be multiple branches of Hopf bifurcation.
In the ANN models, the following behaviour associated with the codimen-sion two points has been observed (Belair et al., 1996; Campbell et al., 2005; Shayer and Campbell, 2000; Yuan and Campbell, 2004): (i) multistability between a periodic solution and one or more equilibrium points; (ii) bistability between two periodic solutions (both synchronous or one synchronous and one asynchronous); and (iii) switching from one stable solutions to another as the delay is changed for a fixed coupling strength or as the coupling strength is changed for a fixed delay. The switching in (iii) may take place through a region of bistability or a region where the trivial solution is stable. Note that situation (i) leads to a different type of oscillator death than that discussed in the previous subsection: a slight perturbation can cause the system to switch from the stable oscillatory solution to the stable equilibrium solution, with no change in the parameter values.
Most of this behaviour has been confirmed in systems with biophysically relevant models for the neurons. In their studies of rings of Fitzhugh-Nagumo oscillators with time delayed gap-junctional or sigmoidal coupling, BuriC and TodoroviC (2003) and Buric et al. (2005) have documented almost all the behaviour observed in the ANN models including switching between different oscillation patterns and bistability between different oscillation patterns. For a system of two van der Pol oscillators with linear delayed coupling (i.e. (3) with no Xj(t) term), Sen and Rand (2003) have numerically observed and Wirkus and Rand (2002) have analytically proven the following sequence as the time delay is increased: in-phase oscillations ^ bistability between in-phase and anti-phase oscillations ^ anti-phase oscillations. They also observed the reverse sequence for different values of the coupling strength. Delay-induced bistability between in-phase oscillations and suppression oscillations (i.e. one cell oscillates and the other is quiescent) has been observed in models of hip-pocampal interneurons (Skinner et al., 2005a,b). Here the delay was synaptic and modelled via an extra equation representing the chemical kinetics of the synapse. Bistability between different types of travelling pulses has been observed in certain integrate-and-fire networks with delayed excitatory synaptic connections (Golomb and Ermentrout, 1999, 2000). In particular, they observe a switch from continuous travelling pulses to lurching travelling pulses as the time delay is increased with a transition region where there is bista-bility between the two types. This behaviour seems to be associated with a subcritical Hopf bifurcation.
Foss et al. (1996) and Milton and Foss (1997) have studied multistability in models for a delayed recurrent neural loop. Their model consists of a single excitatory neuron with delayed inhibitory feedback. They showed that up to three stable oscillatory patterns can coexist and that switching between the attractors can be induced by small perturbations in the neuron voltage (Foss et al., 1996) or by noise (Foss et al., 1997). These results have been replicated in experimental studies of a hybrid neural computer device consisting of an Aplysia motorneuron dynamically clamped to a computer which provides the delayed feedback (Foss and Milton, 2000, 2002). A possible cause of the multistability in these delayed feedback systems maybe period doubling bifurcations (Ikeda and Matsumoto, 1987). Bistability between different oscillation patterns was also observed in preparations of small Aplysia neural circuits (Kleinfeld et al., 1990).
Bifurcation induced transitions between different attractors have been observed in several experiments. In an experimental electrical circuit system, Ramana Reddy et al. (2000) have observed the sequence: in-phase oscillations ^ no oscillations ^ anti-phase oscillations as the time delay in the (gap-junctional) coupling is increased. Transitions from in-phase to anti-phase oscillations have been observed in human bimanual coordination experiments (Kelso et al., 1981; Kelso, 1984; Carson et al., 1994); see also the review article of Jantzen and Kelso (2006). One model which explains these experiments incorporates time delays in the coupling (Haken et al., 1985).
There are several approaches to studying synchronization. I will not review the details here, but give some indication which of these have been extended to delay differential equations and what the results are.
There is a very large literature on synchronization in artificial neural networks, some of which addresses systems with time delays (Campbell et al., 2006; Wu et al., 1999; Yuan and Campbell, 2004; Zhou et al., 2004a,b). Most of these papers use Lyapunov functionals to show that the all solutions synchronize as t ^ to, for appropriate parameter values. Although the equations of the individual elements are not relevant for modelling biophysical neurons, the techniques of analysis may be carried over to neural systems. A common conclusion in many of these papers is that if the strength of the coupling is small enough, one can achieve synchronization for all t > 0. However, synchronization may mean that all elements asymptotically approach the same equilibrium point.
As I have mentioned elsewhere in this chapter, a basic principle of delay differential equations such as (13) is that the behaviour of the system for small delay is often qualitatively similar to that for zero delay. Thus if the neurons are synchronized for a given value of the coupling with zero delay they should remain synchronized for small enough delays in the coupling. Unfortunately, quantifying "small enough" may be difficult and will generally depend on the particular neural model involved. Recall the example illustrated in Fig. 1. We showed that for c > 0 large enough (sufficiently large excitatory coupling) the undelayed system exhibits synchronized oscillations. We expect these oscillations to persist for t > 0 at least until one reaches the first thick Hopf bifurcation curve where synchronous oscillations are destroyed. (If the Hopf bifurcation is subcritical, the oscillations may persist above the curve). Thus, for this particular example, the Hopf bifurcation curve gives a lower bound on "how small" the delay must be to preserve the synchronization found for zero delay. Note that this does not preclude synchronization occurring for larger values of the delay, which is the case in this example. A similar situation is seen for coupled van der Pol oscillators in (Wirkus and Rand, 2002). Another example is the work of Fox et al. (2001) who studied relaxation oscillators with excitatory time delayed coupling. They showed that synchrony achieved for zero delay is preserved for delays up to about 10% of the period of the oscillation, for a variety of different models. The one exception is when the right hand side of the equation is not a differentiable function, in which case synchronization is lost for t > 0. Crook et al. (1997) observed a similar phenomenon for a continuum model of the cortex, with excitatory coupling and distance dependent delays. Namely, they found for small enough delay the synchronous oscillation is stable, but for larger delays this oscillation loses stability to a travelling wave.
More complicated situations occur when both excitatory and inhibitory connections exist. Ermentrout and Kopell (1998); Kopell et al. (2000); Kar-bowski and Kopell (2000) have studied a model for hippocampal networks of excitatory and inhibitory neurons where two types of synchronous oscillation are possible. They show that persistence of the synchronous oscillations with delays depends subtly on the currents present in the cells and the connections present between cells.
So far I have discussed synchronization in spite of delays. I now move on to the more interesting case of synchronization because of delays. This situation can occur when there are inhibitory synaptic connections in the network. This has been extensively documented and studied when the delay is modelled by slow kinetics of the synaptic gating variable (van Vreeswijk et al., 1994; Wang and Buzsaki, 1998; Wang and Rinzel, 1992, 1993; White et al., 1998). Further, Maex and De Schutter (2003) suggest that the type of delay is not important, just the fact that it leads to a separation in time between when the pre-synaptic neuron generates an action potential and the post-synaptic neuron receives it. They confirm this for a network of multi-compartment model neurons with fast synaptic kinetics and a discrete conduction delay. This idea is further supported by the observation of synchronization via discrete delayed inhibition in a number of artificial neural network models (Campbell et al., 2004, 2005; Shayer and Campbell, 2000). Finally we illustrate this with our coupled Fitzhugh-Nagumo model. Consider the part of Fig. 1 with c < 0 (inhibitory coupling). For sufficiently large coupling strength and zero delay the system tends to an asynchronous phase-locked state. This state persists for t > 0 sufficiently small, however, for t large enough a stable synchronous state may be created in the Hopf bifurcation corresponding to the thin curve.
Only a few studies have looked at synchronization with time delayed gap-junctional coupling. One example is the work of Dhamala et al. (2004) which shows that for two gap junctional coupled Hindmarsh-Rose neurons synchronization is achieved for smaller coupling strengths if there is a nonzero time delay in the coupling. Another is the work of Buric et al. (2005).
There are very few results concerning neural systems with distributed delays, thus I will review some general results, mostly from the population biology literature, which should carry over to neural systems. What has emerged from this literature is a general principle that a system with a distribution of delays is inherently more stable than the same system with a discrete delay. Some specific results to support this are described below.
Bernard et al. (2001) analyzed the linear stability of a scalar system with one and two delays in terms of generic properties of the distribution g, such as the mean, variance and skewness. For the uniform and continuous distributions, they have shown that stability regions are larger than those with a discrete delay.
Jirsa and Ding (2004) have analyzed an n x n linear system with linear decay and arbitrary connections with a common delay. They have shown, under some mild assumptions, that the stability region of the trivial solution for any distribution of delays is larger than and contains the stability region for a discrete delay.
Campbell and Ncube (2006) have shown that it is more difficult to get delay induced oscillations with distributions of delays of the form (6) with Tm = 0. For large variance (m =1) delay induced instability is impossible and for smaller variance (m > 1) the mean delay needed for instability is much larger than the discrete delay value. They have also shown that sufficiently small variance in the distribution is needed to get the bifurcation interactions which may lead to multistability, oscillator death and attractor switching discussed above.
Atay (2003a, 2006) has studied the same model as Ramana Reddy et al. (1998) only with distributed delays of the form (6) with g given by (7). He shows it is easier to destroy oscillations with a distribution of delays than with a discrete delay, in the sense that there is a larger region of oscillator death in the parameter space consisting of the mean delay and the strength of the coupling. As the variance of the distribution increases the size of this region increases.
Thiel et al. (2003) studied a scalar equation representing a mean field approximation for a population pyramidal cells with recurrent feedback, first formulated by Mackey and an der Heiden (1984). They show that having a uniform distribution of delays simplifies the dynamics of the system. The size of the stability region of the equilibrium point is larger and larger mean delays are needed to induce oscillations. Complex phenomena such as chaos are less likely to occur, or totally precluded if the variance of the distribution is sufficiently large. The model with a distribution of delays better explains the appearance of periodic bursts of activity when penicillin is added to a hippocampal slice preparation (which reduces the coupling strength).
In this chapter I showed how time delays due to conduction along the axon or dendrite or due to transmission across the synapse could be modelled with delay differential equations. I outlined some of the tools available for analyzing such equations and reviewed some of the literature about such models. Some key observations are:
- Time delays can lead to the creation of type II oscillations, especially in systems with delayed inhibitory coupling.
- Time delays can destroy type II oscillations in a network of intrinsically oscillatory neurons with gap junctional coupling.
- If a system has a stable synchronous oscillation when there is no delay in the coupling, the solution remains stable for small enough delay, but may lose stability for larger delay.
- A system with inhibitory coupling which does not have a stable synchronous oscillation for zero delay, may have one if the delay is large enough.
- Time delays may lead to bistability between different type II oscillatory solutions (e.g. synchronous and anti-phase) or switching between different type II oscillatory solutions.
There are a number of problems which still require further study. These include: determining the effect of delay on the generation and destruction of type I oscillations (infinite period bifurcations), applying and/or extending the methods used to study synchronization in artificial neural networks to biophysical neural networks, and studying the effect of distributions of delays on biophysical neural networks.
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