Adaptive and Phase Selective Spike Timing Dependent Plasticity in Synaptically Coupled Neuronal Oscillators

We consider and analyze the influence of spike-timing dependent plasticity (STDP) on homeostatic states in synaptically coupled neuronal oscillators. In contrast to conventional models of STDP in which spike-timing affects weights of synaptic connections, we consider a model of STDP in which the time lags between pre- and/or post-synaptic spikes change internal state of pre- and/or post-synaptic neurons respectively. The analysis reveals that STDP processes of this type, modeled by a single ordinary differential equation, may ensure efficient, yet coarse, phase-locking of spikes in the system to a given reference phase. Precision of the phase locking, i.e. the amplitude of relative phase deviations from the reference, depends on the values of natural frequencies of oscillators and, additionally, on parameters of the STDP law. These deviations can be optimized by appropriate tuning of gains (i.e. sensitivity to spike-timing mismatches) of the STDP mechanism. However, as we demonstrate, such deviations can not be made arbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights. Thus if accurate phase-locking in the system is required then an additional tuning mechanism is generally needed. We found that adding a very simple adaptation dynamics in the form of slow fluctuations of the base line in the STDP mechanism enables accurate phase tuning in the system with arbitrary high precision. Adaptation operating at a slow time scale may be associated with extracellular matter such as matrix and glia. Thus the findings may suggest a possible role of the latter in regulating synaptic transmission in neuronal circuits.

For the sake of simplicity we consider the case when only one of the two coupled phase oscillators, e.g. postsynaptic, is supplied with the STDP-type mechanism. We will also assume that if the value of z post is fixed (within an interval) then both oscillators operate in a vicinity of asymptotically stable limit cycles. In addition, we will suppose that the values of g syn are negligibly small so that we can investigate stabilizing effects of STDP in the combined system independently from effect of phase pulling due to the instantaneous synaptic coupling between oscillators [1].
Under the assumptions above, phase of the oscillations in V pre , w pre -system (see e.g. [2]) can be defined as a function φ pre (V pre , w pre , ∆I) such that ∂φ pre ∂V preV pre + ∂φ pre ∂w preẇ pre = ω 0 (I pre + ∆I), (S1.1) where ω 0 (I pre + ∆I) is the natural frequency of oscillations. Since the frequency of oscillations decreases with ∆I (see Fig. 13), the function ω 0 (·) is strictly monotone and non-increasing. Moreover, for a technical reason we will assume that the function ω 0 (·) is differentiable or that it can be approximated Notice that if the value of z post is allowed to vary and |ż post | is sufficiently small, thenφ post ω 0 (z post ) provided that |ω 0 (z post )| |ż post |. The latter inequality reflects that the frequency of oscillations is much higher than the time constant of the STDP. In what follows we will consider the case when this asymptotic holds.
Taking (S1.1), (S1.2) into account we can conclude that dynamics of the relative phase, ϕ, is to satisfy the following equationφ = ω 0 (z post ) − ω 0 (I pre + ∆I). Denoting is a continuous, locally Lipschitz, strictly monotone non-decreasing function, and ω ∈ R is the frequency de-tuning parameter, we arrive at the following system of equations: With regards to the values of ω, we will assume that ω ∈ range(f ). It is also clear that ϕ(t pre (i)) = Φ i , i = 1, 2, . . . according to the definition of variable Φ i in (4). Let t ∈ [t post (i), t post (i) + T post (i)), then the last equation in (S1.3) can be explicitly integrated giving rise to Thus taking (S1.3) and (S1.4) into account and approximating the function f (·) by a linear one, f (z post ) = f * + f 0 z post , we can now express ϕ(t) for t ∈ [t post (i), t post (i) + T post (i)) as: Noticing that ϕ(t post (i) + T post (i)) = Φ i+1 , ϕ(t post (i)) = Φ i , denoting z i = z post (t post (i)), z i+1 = z post (t post (i) + T post (i)), and using the fact that ϕ(t) is continuous in t, we arrive at Let Φ * , z * be an equilibrium of (S1.5). Given that G(·) is differentiable, and neglecting dependence of T post (i) on z post if z post is close enough to z * , we can linearize the dynamics of (S1.5) about Φ * , z * : (S1.7) Denoting δΦ = Φ * − Φ c (cf. (10)) we rewrite (S1.6) as where, according to the second equation in (S1.5), the value of δΦ is Let σ 1 , σ 2 be eigenvalues of the matrix Then, provided that the fluctuations of T post (i) are sufficiently small, condition |σ 1 | < 1, |σ 2 | < 1 ensures that the fixed point Φ * , z * is stable in the sense of Lyapunov. The eigenvalues of K i can be expressed as (S1.9) According to (S1.7), (S1.9), checking if |σ 1 | < 1, |σ 2 | < 1 holds requires availability of the estimates/values of f 0 and G 0 . The value of f 0 can be explicitly inferred from Fig According to the figure, there is a range of values of k post for which both |σ 1 | and |σ 2 | are less than one, and hence the fixed point is stable. The boundaries of this range are largely consistent with numerical analysis of the original system summarized in the bifurcation diagram in Fig. 11 (left panel). Minor disagreements between the figures can be observed for k post small. These, however, are due to the following two factors. First, the fixed point itself disappears when the values of k post become sufficiently small. Second, model (S1.3) does not include the influence of synaptic coupling, I syn . Neither of these factors are accounted for in expression (S1.9), and that is why the stability diagram in Fig. 11 (right panel) is inconsistent with the bifurcation one (left panel) for k post small.
Summarizing the analysis above one can conclude that • if f 0 , G 0 > 0 then, for a broad range of T post , there will always exist values of the STDP parameters, k post , α post , such that the fixed point of (S1.5) is locally exponentially stable for these values; • if the values of k post are made large enough, i.e. when max{|σ 1 |, |σ 2 |} > 1 (which is always possible to achieve, see (S1.7), (S1.9)), the corresponding fixed point becomes unstable.
• on the other hand, if the values of k post are too small then the fixed point ceases to exist.
An alternative strategy for assessing stability of the equilibria of (S1.3) can be carried out without explicit integration of the second equation in (S1.3). If k post ∈ R >0 , α post ∈ R >0 are sufficiently small then solutions of (S1.3) can be approximated by that oḟ (S1.10) It is clear that equilibria of (S1.10) can be determined from where G −1 is, in general, a set-valued function. If Φ * is such that ∂G/∂ϕ(Φ * − Φ c ) > 0 then one can conclude that Φ * , z * is an asymptotically stable equilibrium of (S1.3). The conclusion follows from the analysis of the time-derivative of the following Lyapunov candidate: followed by invoking the Barbalatt's lemma for demonstrating asymptotic convergence of ϕ(t) to Φ * . An important consequence of the stability analysis above, specifically (S1.3) -(S1.8), is that if λ post is allowed to vary then, subject to the choice of k post , α post , the dynamics of (1) in which z post evolves in accordance with (6) locally satisfies the following constraint: where β(·) is a strictly monotone, positive, and non-increasing function vanishing asymptotically at infinity, and c is a non-negative constant.
Local non-uniform small gain theorem Consider a system with input, u, and let the evolution of its state, x, be governed bẏ where the function f : R n × R × R → R n is continuous in x, u, t and bounded in t, and u : R → R is a continuous function. For the sake of notational compactness we denote z(τ ) ∞, [a,b] [t0,t] ≤ ∆ u }, and suppose that for any t 0 ∈ R, t > t 0 and all (x 0 , u) ∈ Ω(t 0 , t), the solutions of (S1.17) satisfying x(t 0 ) = x 0 are defined, and the following holds where β(T ) is a strictly monotone continuous function: lim t→∞ β(t) = 0, β(0) ≥ 1. Let us suppose that for all t 0 input u in (S1.17) is evolving according to Then the following statement holds for interconnection (S1.17), (S1.18) (cf. [3], [4]) Proposition 1 Consider interconnection (S1.17), (S1.18), and suppose that the domain is not empty for some d < 1, κ > 1. Let Then for all (x 0 , u(t 0 )) ∈ Ω γ ∩Ω ∆ the state (x(t), u(t)) of the interconnection is bounded. Furthermore, if there is a function w : R n → R ≥0 such that (S1.20) then for every divergent and ordered sequence {t i }, i = 0, 1, . . . , t i < t i+1 , the following holds: Proof of Proposition 1. Let us introduce a strictly decreasing sequence {σ i }, i = 0, 1, . . . , such that σ 0 = 1, and σ i asymptotically converge to zero. Let {t i }, t i , i = 1, . . . be an ordered infinite sequence of time instances such that u(t i ) = σ i u(t 0 ).
In case this equality does not hold, nothing remains to be proven since u(t) will always be bounded and separated away from zero for t ≥ t 0 .
We wish to show that the amount of time needed to reach the set {(x, u)| u = 0} from the given initial condition is larger than any positive number, i.e. infinite.
Consider time differences T i = t i − t i−1 . It is clear that: (S1.22) In addition to {t i } we introduce another auxiliary sequence {τ i }, τ i = τ * , τ * ∈ R >0 , i = 1, . . . . Given that the partial sums i τ i = i τ * diverge we can conclude that proving the implication will automatically assure that x(t), u(t) are bounded for all t ≥ t 0 . We prove (S1.23) by induction wrt i.
Finally, let us show that (S1.20) implies (S1.21). Suppose that this is not the case, and for an ordered diverging sequence {t i } there is a δ > 0 such that w(x(t )) > δ, ∀ t ∈ [t i−1 , t i ], i = 1, 2, . . . . Hence (S1.27) According to the first part of the proposition u(t i ) is bounded for all i. On the other hand, using (S1.27) one can conclude that for any given arbitrarily large M , there is an n > 0 such that u(t n ) ≤ −M . Thus we have reached contradiction which proves (S1.21) .