A Model of Fast Hebbian Spike Latency Normalization

Hebbian changes of excitatory synapses are driven by and enhance correlations between pre- and postsynaptic neuronal activations, forming a positive feedback loop that can lead to instability in simulated neural networks. Because Hebbian learning may occur on time scales of seconds to minutes, it is conjectured that some form of fast stabilization of neural firing is necessary to avoid runaway of excitation, but both the theoretical underpinning and the biological implementation for such homeostatic mechanism are to be fully investigated. Supported by analytical and computational arguments, we show that a Hebbian spike-timing-dependent metaplasticity rule, accounts for inherently-stable, quick tuning of the total input weight of a single neuron in the general scenario of asynchronous neural firing characterized by UP and DOWN states of activity.


A.1 Theoretical tools
To analyze the model without leak, we use Bernoulli and binomially distributed random variables. A Bernoulli random variable X takes the value 1 with probability p and 0 with probability 1 − p for 0 ≤ p ≤ 1. Denote this by X ∼ Be(p). A binomial random variable Y , with parameters n and p, is a sum of n independent Bernoulli variables with success probability p. Denote this by Y ∼ Bi(n, p). Bernoulli random variables are used to model the type of learning signal a synapse receives. The binomial distribution is used in two contexts, to sum up the Bernoulli variables in the memory from the model without leak and to sum up the total input weight in a volley when synapses are probabilistic.
Let X 1 ∼ Be(p 1 ), . . . , X n ∼ Be(p n ) be independent Bernoulli distributed random variables, and let X = n i=1 X i . Then, the following inequalities hold for ε ≥ 0, and µ ≡ E[X] = n i=1 p i .
Theorem 2 (Variable Drift [1, Theorem 4.6]). Let (X t ) t≥0 be a sequence of random variables over a state space 0 ∈ S ⊆ R + 0 such that s min ≡ inf{x ∈ S | x > 0} > 0. Assume further that X 0 = s 0 for some s 0 ∈ S. Let T be the random variable that denotes the earliest point in time t ≥ 0 such that X t = 0. Suppose there is an increasing function h : then, for all x, A.2 Scaling of the memory trace with synaptic parameters In this section, the exact derivation of how to choose the memory size as in Equation (19) such that at most a δ fraction of the synapses change their weight in the stable state is given. Recall that in the model without leak for ε > 0, and some value of d s , for which one wants an upper bound on stability, the threshold parameters are set as in Equation (18). The requirement from above on limiting weight changes in the stable state translates to the following two inequalities: By Theorem 1 To see how this upper bound scales with the size of the stable region for an absolute deviation from d s , we set ε = x/d s where x is a positive integer.

A.3 Convergence
We derive an upper bound for the expected number of weight updates until the input weight is close to the stable state. Assume that initially d s > (1 + 2ε)d * s and denote by T the number of weight updates until Let ε < 1/4 be the same as in Equation (18) for θ P and θ D . Assume that such that potentiation is balanced with excitation in the stable state. For simplicity, assume that the synapses are deterministic (p r = 1). Furthermore, assume that the synapse is strong, i.e., w vu = 1. For convenience, define µ 1 ≡ θv ds M . The probability of the event m(t) ≤ θ D can be lower bounded as follows: where in (1) where in (1) the inequality ds+1 d * s +1 ≥ 1 + ε was used and in (2) Theorem 1 was used. Assuming d s ≥ (1 + 2ε)d * s , the expected absolute weight drift, the difference of Equations (15) and (16), can be lower bounded as follows: If d s ≥ d * s , then p w→s was chosen such that (d − d s )p w→s = O(d s p s→w ). By increasing M , f (ε, M ) can be made arbitrarily close to 1 and g(ε, M ) arbitrarily close to 0. Therefore, by choosing one can lower bound the amount of progress in a weight update toward the stable state by Denote by d s (t) the number of strong synapses after t learning rounds and set d s (t) = max{1, d s (0) − (1 + 2ε)d * s }. Set T = min t {t | d s (t) = 0} which is the time needed to hit the interval around d * s in the shifted process, d s . By Theorem 2, the variable drift theorem, The case for d s = (1 − 2ε)d * s follows an analogous argument.
A.4 Validity of the homeostatic mechanism regardless of the random order assumption In this section, we prove that the intrinsic homeostasis mechanism works for any probability distribution on the set of all orderings. More precisely, if in each volley an ordering is drawn from this distribution, then the input weight remains bounded. This result shows that the mechanism provides negative feedback for arbitrary input distributions. This result essentially follows from the fact that only a few synapses can remain strong because only θ v of them are needed to activate the target neuron. For large input weight, the expected weight drift is negative. Denote a synapse between neuron u and v by (u, v), and denote their weight by w uv . In what follows, d * s denotes as usual the stable state for the standard process when the input order is chosen uniformly at random.
For a set S, denote by |S| the number of elements it contains. To study this setting, let E 1 ≡ {(u, v)|w uv = 1} be the set of strong synapses and for e uv ∈ E 1 , let X uv be an indicator random variable for the event that u spikes before v in this volley (X uv is 1 if the event occurs and 0 otherwise). Furthermore, let p uv = Pr[X uv = 1] be the success probability. Then, by linearity of expectation and the fact that only θ v of the strong synapses are early, For 0 < ε < 1/2, assume d s > d * s 1−2ε , i.e., the input weight, is too large. Strong synapses with p uv > (1 − 2ε) θv d * s = p * have a chance to retain their weight whereas other strong ones will have m(t) < θ D with high probability by Theorem 1. To formalize this, denote by E fast ⊆ E 1 the synapses that have p uv > p * and by E slow ⊆ E 1 those who have p uv ≤ p * . Furthermore, let p fast ≡ |E fast | −1 (u,v)∈E fast p uv and p slow ≡ |E slow | −1 (u,v)∈E slow p uv so that by Equation (37) it follows that Using p * < p fast , one obtains the following upper bound on the number of fast synapses: Take the derivative w.r.t. p slow to maximize the upper bound on |E fast |; that is, d dp slow For d s > θv p * = d * s 1−2ε , the upper bound for |E fast | is maximized for p slow = 0. Therefore, Since d s > d * s 1−2ε , there is high probability of at least d s − d * s 1−2ε slow synapses that will turn weak with probability p s→w in the next weight update. Once at most d * s synapses (and at least θ v ) among the fastest ones are strong, the weight cannot increase further and it stays upper bounded with high probability.