User:Nancy Kopell/Proposed/Fast threshold modulation

• Dr. Nancy Kopell, Boston University, Boston MA

• Dr. Enno de Lange, Department of Physiology, University of Bern, Bern, Switzerland

Figure 1: Two FitzHugh-Nagumo neurons coupled by fast threshold modulation. The red neuron is stimulated at the red arrow to elicit a spike. When it exceeds the red dotted coupling threshold, the coupling to the blue neuron becomes active.

Fast threshold modulation (FTM) is a way of coupling dynamical systems. It can be seen as a model for a synapse and therefore is often used in the context of neuron modeling.

In FTM the coupling signal switches between two values, often an "on" and "off" value. This switching is governed by a variable of the coupling system, the membrane potential of the pre-synaptic cell in the context of neuronal modeling. When this variable exceeds a predefined threshold the coupling signal switches on.

Fast threshold modulation and its synchronization properties were first described by Somers and Kopell (1993).

Use of FTM

Fast threshold modulation is mainly used to couple neuron models of two to four ordinary differential equations and, more specifically, when the models are relaxation oscillators. The movement on two distinct timescales, typical of relaxation oscillators, is used in the mathematical analysis of the synchronization properties of FTM. Consider two 2-dimensional neuron models, labeled \(i\) and \(j\ ,\) with membrane potential \(v\) and slow adaptation current \(w\ .\) The movement on two distinct timescales is made explicit by a small parameter \(\epsilon \ll 1\ .\) Neuron \(i\) then couples to neuron \(j\) in the following way: \[ \begin{array}{rcl} \displaystyle \epsilon\dot{v_i} & = &\displaystyle f(v_i,w_i,I_{syn}^{(i)})\\[0.5em] \displaystyle \dot{w_i} & = &\displaystyle g(v_i,w_i)\\[1em] \displaystyle \epsilon\dot{v_j} & = &\displaystyle f(v_j,w_j,I_{syn}^{(j)})\\[0.5em] \displaystyle \dot{w_j} & = &\displaystyle g(v_j,w_j) \end{array} \]

Figure 2: A sigmoidal function for FTM. The value changes between \(I^-\) and \(I^+\) with a steep ramp as a transition. The gray area indicates the values of the presynaptic membrane potential for which the transition occurs.

Here the coupling current is \(I_{syn}^{ {j} }=\Theta^{(j)}(v_i)\ .\) The coupling function \(\Theta\) is a sigmoidal function of the presynaptic membrane potential (see Figure 2) or, in the limiting case, a Heaviside step function. This introduces a (soft) threshold \(v_i=\theta\ :\) below a certain presynaptic membrane potential the coupling is (very close to) \(I^-\) and above this threshold, the coupling is (very close to) \(I^+\ .\) For inhibitory coupling this behavior can be inverted.

An example of a coupling function that is often used is the hyperbolic tangent. The coupling current of neuron \(j\) with presynaptic neuron \(i\ ,\) coupling threshold \(\theta\) and coupling strength \(g_{syn}\) can then be modeled as \[ I_{syn}^{(j)}(v_i) = g_{syn}\left(\frac{1}{2}\tanh \left(\lambda(v_i-\theta)+1\right)\right) \] where parameter \(\lambda\) changes the steepness of the ramp of the sigmoidal. For a conductance-based version, the current is \[ I_{syn}^{(j)}(v_i,v_j) = -g_{syn}\left(\frac{1}{2}\tanh \left(\lambda(v_i-\theta)+1\right)\right)\left(v_j-v_{syn}\right) \] with \(v_{syn}\) the synaptic reversal potential. For excitatory coupling \(v_{syn}\) is higher than the range of voltage values for the oscillation, and for inhibitory coupling, \(v_{syn}\) lies below that range.

Modeling assumptions

Fast threshold modulation is a memory-less coupling so an important modeling assumption is that the dynamics of the coupling which is modeled can be neglected. In a neuronal context this means that the dynamics of the synapse is fast with respect to the intrinsic timescales of the neurons.

For many analytical studies of FTM three assumptions are made about the coupling function \(f(v_i)\) and the dynamics of the neuron models. First the neurons are expected to move on two distinct timescales, as is the case for relaxation oscillators such as the FitzHugh-Nagumo and Morris-Lecar model. Second the ramp of the sigmoidal is steep with respect to the membrane potential. This guarantees the threshold-like behavior of the coupling. Third the ramp lies within the fast transition of the membrane potential.

The discussion below focuses on two-dimensional, spiking, models, but the analysis is also valid for bursting neuron models, provided that the synapse is fast compared to the times on the upper and lower – burst – plateau (Kopell & Somers, 1995; Kopell & Ermentrout, 2000; Belykh et al., 2005).

FTM in one-way coupling

To understand the mechanism behind fast threshold modulation it is useful to study the interaction of pairs of coupled model neurons. If only one of the two neurons receives a coupling signal the coupling is one-way and a (coupled) post and (coupling) presynaptic neuron can be distinguished.

Figure 3: Impulse detection. FTM causes an all-or-nothing response in two one-way coupled FitzHugh-Nagumo neurons. FN neurons have standard parameter values, except \(\epsilon=0.01\ .\) Coupling function is \(I_{syn}(v_{i})=g_{syn}(1/2(\tanh(10*(v_{i})+1))\ .\)

Impulse Detection

When the postsynaptic neuron is quiescent, it can react on incoming pulses from the presynaptic cell if the coupling signal is strong enough. Figure 3 illustrates the principle of impulse detection. A pulse in the presynaptic neuron causes a threshold crossing of the coupling function (dashed line). The coupling current of the postsynaptic neuron is activated, which gives an impulse, generating a spike.

In FTM-coupled relaxation oscillators, there is a clear threshold value for the coupling strength \(g_{syn}\) necessary to generate a spike: it exhibits all or nothing behavior. The value if the threshold depends on the characteristics of the neuron model.

Figure 4: Phase locking. Pre- and postsynaptic neurons both spike with different initial phase and frequency. The synchronization is not full and a slight phase lag remains in the asymptotic solution (inset). Models as in Figure 3, but \(I=0.9\) for presynaptic neuron and \(I=0.8\) for postsynaptic neuron. Coupling strength \(g_{syn}=0.1\ .\)

Phase locking

When both presynaptic and postsynaptic neurons are oscillating, possibly with different frequency, coupling can make them synchronize their frequencies and let their phases approach to a small difference. This is known as phase locking. Unlike the case of mutual coupling discussed below, the postsynaptic neuron will always lag the presynaptic neuron. As for impulse detection there is a minimum coupling strength for which phase locking occurs. Typically this minimum coupling strength is lower than that for impulse detection in the same neuron model.

In Figure 4, an example is shown with two FitzHugh-Nagumo oscillators. Although they start with opposite phase and have different frequency of oscillation, the postsynaptic cell synchronizes to the phase and frequency of the presynaptic cell within eight periods of oscillation.

FTM in two-way coupling

Two-way coupling refers to the situation in which there are two coupling currents, with each cell pre-synaptic in one of the currents. An extensive treatment of the synchronization properties of two-way FTM coupled relaxation oscillators has been given by Somers and Kopell in a series of papers in the nineteen nineties [Somers & Kopell, 1993; Kopell & Somers, 1995; Somers & Kopell, 1995].

Figure 5: Time as a metric. Two identical FN oscillators move along the limit cycle with the red oscillator leading by a fixed time difference. The phase of both the oscillators on the limit cycle (in grey) is plotted for the point where the red oscillators just reaches the knee point (circles) and 150 time steps later (crosses)

Time as a metric

To think about synchronization, we define a convenient metric to measure the distance at a given time between two oscillators whose trajectories travel along the same slow curves. The metric is the time it takes to go from the position of one oscillator to the other along that curve. Figure 5 illustrates how the time metric can be different from a (Euclidian) space metric for relaxation oscillators: although the oscillators have the same phase as defined by the time metric for the two points shown in the figure, the distance in state-space between both oscillators is very different for the two points.

Compression across a jump

It is clear that if the time difference between two oscillators always decreases they will synchronize. In relaxation oscillators, the time difference between the oscillators may change instantly on the rapid jump between the nullclines. The amount with which the time difference changes on each jump is called compression.

Figure 6: Limiting synchronous solution. The limiting synchronous solution of two-way coupled relaxation oscillators is shown in red in relation to the nullclines of the oscillator for the two possible values of the coupling current.

Compression across a jump

The formal proof of synchronization considers two identical ideal relaxation oscillators. The oscillators move on the outer, stable, branches of the cubic nullcline and when arriving at the knee points, \(k_L\) and \(k_R\ ,\) they jump instantaneously to the other branch. In this situation, the transition of the coupling threshold \(\theta\ ,\) which lies within this jump, occurs instantaneously.

A transition of the coupling threshold of one oscillator changes the cubic nullcline of the other. When both oscillators are synchronized this change occurs simultaneously for both oscillators. So both oscillators move on one branch of the "coupling off" nullcline, but on the other branch they move on the, different, "coupling on" nullcline. This trajectory is known as the 'limiting synchronous solution' (LSS) (see Figure 6).

Figure 7: Illustration of the requirements for the existence of an asymptotically stable synchronous solution in two-way FTM coupled relaxation oscillators.

These fast jumps and the modulation around the coupling threshold alone do not necessarily produce an asymptotically stable synchronous solution. There is one additional condition, central to the proof of stable synchronization: Suppose that two oscillators are both on the left branch of the nullcline and that when the neuron closest to the knee point \(k_L\) - the leading neuron - arrives at said knee point, the other neuron is so close to the knee point that it will jump directly to the right branch and not to the left branch of the "coupling on" nullcline. This situation is shown in Figure 7. Let \(j(\cdot)\) indicate the arrival point on the right branch of the LSS of a point on the left branch after the jump. In the case of ideal relaxation oscillators this means that for a point \(p\) on the left branch, \(j(p)\) is the point on the right branch with the same value of the slow variable as \(p\ .\) Now let \(\tau{}(p)\) be the time from \(p\) to \(k_L\) over the left branch of the LSS and \(\tau{}(j(p))\) the time from \(j(k_L)\) to \(j(p)\) over the right branch. The '’compression’' of \(p\) is then defined by \(C(p)\equiv \tau{}(j(p))/\tau{}(p)\ .\)

Essential to understand the idea behind compression is to think about the distance between the two oscillators in terms of time. If the compression is smaller than one the after the jump the oscillators are closer to each other in time on the trajectory than before. It is easy to see intuitively how this situation eventually leads to synchrony. This approach circumvents the problem that for relaxation oscillators it is hard to define a phase, which is the usual approach in synchronization proofs. The phase is defined as the time evolution of the LSS.

Figure 8: Synchronization and initial conditions for two-way FTM coupled FitzHugh-Nagumo neurons. Lower plot shows the number of periods the neurons need to synchronize (get closer than 1% of their period) for different initial conditions. First neuron always starts in \(k_L\ ,\) indicated by a green cross. Second neuron starts on different points lying on its uncoupled periodic orbit. These points are indicated by colored dots. The red-to-yellow graded dots correspond to asymptotically stable synchronous solutions. The blue dots are either anti-phase solutions, or solutions that synchronize in more than 10 periods. Upper plot shows an example of the membrane potentials of both neurons for initial conditions leading to synchrony within three periods. Parameters of models as in Figure 3, but \(g_{syn}=0.05\) and \(I=0.3\) for both neurons.

Conditions for synchronization, identical oscillators with excitatory coupling

Combining the conditions outlined in the previous section, the following result for asymptotic stability of the limiting synchronous solution of two identical relaxation oscillators can be formulated:

If two oscillators are close enough so that the lagging one jumps immediately to the right branch - the situation pictured in Figure 7 - and if the compression is smaller than one, then the LSS is asymptotically stable.

The proof can be extended to also include situations where the lagging oscillator is higher on the left branch, and for non-ideal relaxation oscillators (\(0 < \epsilon \ll 1\)), but the analysis is more complex. The full details of the proof can be found in Somers and Kopell (1993).

In practice most neuron models have a synchronous solution with a domain of initial conditions leading to this solution which is comprises a large part of the state space. An example of such a situation leading to stable synchrony in two FitzHugh-Nagumo oscillators with \(\epsilon = 0.01\) is shown in Figure 8. The FitzHugh-Nagumo model has a relatively large compression and it synchronizes within three periods; two Morris-Lecar neurons, with a compression of about 0.1, would synchronize even faster.

Computer simulations of arrays of oscillators show that identical relaxation oscillators coupled by FTM synchronize much more rapidly than linearly coupled oscillators. The minimum coupling strength needed to synchronize the array is largely independent of the number of oscillators in the array when FTM is used. This in contrast to linearly coupled arrays, where the coupling has to be increased proportionally to the number of oscillators to achieve synchrony (Somers & Kopell, 1993; Belykh et al., 2005).

Synchronization of non-identical oscillators

If the oscillators are not identical the complete synchronous solution, that is, the solution where both systems move synchronously on identical trajectories, does not exist anymore. However, Somers & Kopell (1995) show that when the frequency difference between two oscillators is not too large, they can still synchronize the jumps between the nullclines. This is achieved by an adaptation of the amplitude of each neuron. Chains of FTM-coupled oscillators can have fully synchronized jumps if the gradient of frequency differences is not too large. This behavior is in contrast with that of phase oscillators, interacting through phase differences. Two phase oscillators with non-identical frequency will never fully synchronize their phase (Kopell & Ermentrout,1986).

Figure 9: Anti-phase synchronization. Two mutually coupled FitzHugh-Nagumo neurons, both starting on the left branch of the nullcline, but one at the upper part of the branch and the other near the knee, (case 1d; see text,) show stable anti-phase synchronization. Parameters as in Figure 8, but with different initial conditions

Anti-phase solutions

It is possible for two excitatory FTM-coupled relaxation oscillators to have a stable anti-phase solution as shown in Figure 9. An additional requirement is necessary to guarantee the existence of this stable anti-phase solution: the movement on one of the two branches of the nullcline has to be significantly slower than that on the other.

The proof of the existence of this anti-phase solution centers around the existence of initial conditions for which one of the oscillators stays on the same branch for the entire evolution on the other branch of the other oscillator. It uses the fact that the fast transitions of the oscillators and the shape of the coupling function allow for the oscillators to be treated as partially uncoupled. For full details of the proof, see Kopell & Somers (1995).

As shown in Figure 8 and Figure 9, the FitzHugh-Nagumo oscillator has a set of initial conditions which lead to stable anti-synchrony for \(\epsilon =0.01\ .\) For \(\epsilon =0.08\) however, the anti-synchronous solution is not stable anymore. Furthermore the existence depends on the value of \(I\ ,\) which changes the speed on the branches of the cubic nullcline.

References

• Kopell, N & Ermentrout, G.B. (1986) " Symmetry and phase-locking in chains of weakly coupled oscillators" Comm. Pure Appl. Math. 39:623
• Somers, D. & Kopell, N. (1993) "Rapid synchronization through fast threshold modulation" Biol. Cybern. 68:393-407
• Kopell, N & Somers, D. (1995) "Anti-phase solutions in relaxation oscillators coupled through excitatory interactions" J. Math. Biol. 33:261-280
• Somers, D. & Kopell, N. (1995) "Waves and synchrony in networks of oscillators of relaxation and non-relaxation type" Physica D 89:169-183
• Kopell, N. & Ermentrout, G.B. (2000) "Mechanisms of Phase-Locking and Frequency control in Pairs of coupled Neural Oscillators" In: Handbook of Dynamical Systems, vol 3: Toward Applications
• Rubin J. & Terman, D. (2000) "Analysis of clustered firing patterns in synaptically coupled networks of oscillators" J. Math. Biol. 41(6):513-45
• Izhikevich, E. (2000) "Phase Equations For Relaxation Oscillators" SIAM J. Appl. Math. 60:1789-1805
• Belykh, I.V. et al. (2005), "Synchronization of Bursting Neurons: What Matters in the Network Topology" Phys. Rev. Lett. 94(18):188101

See also

Pulse coupling, Synapses, Synchronization, Burst Synchronization, Fitzhugh-Nagumo Model, Hindmarsh-Rose Model, Morris-Lecar Model

External links

Figure source files. MATLAB scripts to generate the figures used in this article.