Title: Spike-timing-dependent Hebbian learning as noisy gradient descent
Venue: NeurIPS 2025
Accepted: yes
Keywords: Biological neural networks

Hebbian learning

Spike-timing-dependent plasticity

Noisy gradient descent

Mirror descent
Abstract: Hebbian learning is a key principle underlying learning in biological neural networks. We relate a Hebbian spike-timing-dependent plasticity rule to noisy gradient descent with respect to a non-convex loss function on the probability simplex. Despite the constant injection of noise and the non-convexity of the underlying optimization problem, one can rigorously prove that the considered Hebbian learning dynamic identifies the presynaptic neuron with the highest activity and that the convergence is exponentially fast in the number of iterations. This is non-standard and surprising as typically noisy gradient descent with fixed noise level only converges to a stationary regime where the noise causes the dynamic to fluctuate around a minimiser.

Spike-timing-dependent Hebbian learning as noisy gradient descent

Abstract

Hebbian learning is a key principle underlying learning in biological neural networks. We relate a Hebbian spike-timing-dependent plasticity rule to noisy gradient descent with respect to a non-convex loss function on the probability simplex. Despite the constant injection of noise and the non-convexity of the underlying optimization problem, one can rigorously prove that the considered Hebbian learning dynamic identifies the presynaptic neuron with the highest activity and that the convergence is exponentially fast in the number of iterations. This is non-standard and surprising as typically noisy gradient descent with fixed noise level only converges to a stationary regime where the noise causes the dynamic to fluctuate around a minimiser.

1 Introduction

Hebbian learning is a fundamental concept in computational neuroscience, dating back to Hebb [16]. In this work, we provide a rigorous analysis of a Hebbian spike-timing-dependent plasticity (STDP) rule. Those are learning rules for the synaptic strength parameters that only depend on the spike times of the involved neurons. More precisely, we consider a neural network composed of d presynaptic/input neurons, which are connected to one postsynaptic/output neuron. The presynaptic neurons communicate with the postsynaptic neuron by sending spike sequences, the so-called spike-trains. Reweighted by synaptic strength parameters w 1 , . . . , w d ≥ 0 , they contribute to the postsynaptic membrane potential. Whenever the postsynaptic membrane potential exceeds a threshold, the postsynaptic neuron emits a spike, and the membrane potential is reset to zero. Experiments have shown the following stylised facts, which lie at the core of Hebbian learning based on spikes: (1) Locality: The change of the synaptic weight w i depends only on the spike-train of neuron i and the postsynaptic spike-train. (2) Spike-timing: The change of the synaptic weight w i depends on the relative timing of presynaptic spikes of neuron i and

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of the postsynaptic neuron. More precisely, a pre-post spike sequence tends to increase w i , whereas a post-pre sequence tends to decrease w i . We refer to Morrison et al. [28] for a more comprehensive list of experimental results on STDP rules.

Hebbian learning rules are well-studied if instead of the precise timings of pre- and postsynaptic spikes, only the mean firing rates are taken into account. These rate-based models exhibit many desirable properties, including performing streaming PCA [23, 19] and receptive field development [13, Section 11.1.4]. Much less is known if the precise timing of pre- and postsynaptic spikes is considered, since the intrinsic randomness of the dynamics complicates the mathematical analysis.

Our main contribution lies in connecting STDP to noisy gradient descent and providing a rigorous convergence analysis of the noisy learning scheme. To this end, we introduce a learning rule for the weights w 1 , . . . , w d , which captures the locality and spike-time dependence of Hebbian STDP. We rewrite the learning rule as a noisy gradient descent scheme with respect to a suitable loss function. The connection to noisy gradient descent and stochastic approximation [24, 34] paves the way for applying mathematical tools from stochastic process theory to analyse the STDP rule. Our analysis of STDP is inspired by the work on noisy gradient descent for non-convex loss functions of Mertikopoulos et al. [27]. By refining their arguments and carefully tracking the error terms, we show an exponentially fast alignment of the output neuron with the input neuron of the highest mean firing rate on an event of high probability. The specialisation of the output neuron to the input neuron of the highest intensity is related to the winner-take-all mechanism in decision making [12, 49, 31, 25, 40]. The competitive nature of Hebbian STDP has been observed by [39, 38, 15] and the specialisation to few input neurons is important for receptive field development [9]. By connecting Hebbian STDP to noisy gradient descent, we are able to provide a mathematical analysis beyond ensemble averages and to quantify the speed of convergence.

Taking into account the intrinsic geometry of the probability simplex, we also relate our learning rule to noisy mirror descent, more precisely to noisy entropic gradient descent, which has been proposed for brain-like learning by Neumann et al. [30], Cornford et al. [10].

The key contributions are:

STDP as noisy gradient descent. We deduce a new framework, in which Hebbian STDP is interpreted as noisy gradient descent. This connection allows us to employ powerful tools from the theory of stochastic processes for analysing Hebbian STDP.

Linear convergence. We prove the alignment of the output neuron with the input neuron of highest intensity at exponential rate on an event of high probability.

Connection to noisy mirror descent. We relate our learning rule to noisy mirror descent, more specifically to entropic gradient descent. This connection facilitates the integration of techniques from both areas, potentially leading to future synergistic effects.

Related literature

Common approaches to understanding STDP restrict to the mean behaviour after taking the ensemble average, e.g. [13, 22, 15], or compute the full distribution using the master equation of the Markov process [13, Section 11.2.4]. Unfortunately, the latter is only feasible in specific scenarios. In [21], the authors consider a general noisy spike-time dependent dynamic which is transformed into a deterministic ODE by imposing a slow learning rate and using the self-averaging effect of the system. A stability analysis reveals structure formation and output stabilisation. One major difference to our work is the influence of the noise. In [21], the variance of the weights grows linearly and a careful comparison of time scales is required. In our work, despite a constant injection of noise into the system, the dynamic for the spike-triggering probabilities converges to a deterministic limit. Secondly, the use of recent ideas from the analysis of noisy SGD allows us to track the influence of the realised noise in every step. A considerable number of previous works derived STDP rules based on the minimisation of a loss function, typically corresponding to the minimization/maximization of some notion of energy or information, see [8, 6, 7, 44, 42, 32, 43, 36]. While this approach is appealing, the mathematical analysis of these learning rules is challenging due to the modifications required to achieve biological plausibility. In contrast, we start with a biologically plausible learning rule and utilise the arising loss function to derive mathematical convergence guarantees of the learning rule. The importance of the choice of a suitable metric for the derivation of the learning rule is laid out in [41]. We refer to [1, 48, 37, 14, 45] and the references therein for further results on STDP. In [20], a related learning rule is mathematically analyzed by relating it to expert aggregation.

1.1 Notation

Linear algebra. For a positive integer d , we write [ d ] := { 1 , . . . , d } and 1 := (1 , . . . , 1) ⊤ ∈ R d . For i ∈ [ d ] we denote by e i the i th standard basis vector of R d . The Hadamard product between two vectors a , b ∈ R d is denoted by a glyph[circledot] b := ( a 1 b 1 , . . . , a d b d ) ⊤ ∈ R d . We write I ∈ R d × d for the identity matrix on R d and ‖ u ‖ 2 = ∑ d i =1 u 2 i for the squared Euclidean norm of a vector u ∈ R d .

Probability. M (1 , p ) denotes the multinomial distribution with one trial ( n = 1 ) and probability vector p = ( p 1 , . . . , p d ) ⊤ , that is ξ ∼ M (1 , p ) if only only if P ( ξ = i ) = p i for any i ∈ [ d ] . We denote by

the probability simplex in R d . We denote by /x31 A the indicator function of a set A .

1.2 Hebbian inspired learning rule

Inspired by Hebbian learning, we consider an unsupervised learning dynamic with d input (or presynaptic) neurons and one output (or postsynaptic) neuron. The i th input neuron has a mean firing rate λ i > 0 describing the expected number of spikes per time unit. The vector λ = ( λ 1 , . . . , λ d ) contains the d mean firing rates. The strength of the connection between the i th input neuron and the output neuron is modulated by the weight parameter w i ≥ 0 , and changes to encode the information of the input firing rates.

We introduce a Hebbian STDP rule in Subsection 2.3 and show that, under some assumptions on the spike-trains, it is equivalent to the following dynamics. If w (0) = ( w 1 (0) , . . . , w d (0)) ⊤ are the d weights at initialisation, the updating rule from w ( k ) to w ( k +1) is given by

w ( k +1) = w ( k ) glyph[circledot] ( 1 + α ( B ( k ) + Z ( k )) ) , (1) where α > 0 is the learning rate and k = 0 , 1 , . . . denotes the postsynaptic spike time. The d -dimensional vector B ( k ) is the standard basis vector pointing to the presynaptic neuron, which triggered the ( k +1) st postsynaptic spike. It is given as B ( k ) = ∑ d i =1 /x31 ζ k = i e i , the one-hot encoding of independent multinomial random variables ζ k ∼ M (1 , p ( k )) , with k -dependent probability vector

Since the probabilities p i ( k ) model the probability that the ( k +1) st postsynaptic spike is triggered by neuron i = 1 , . . . , d , we call them (postsynaptic) spike-triggering-probabilities . The i.i.d. d -dimensional vectors Z ( k ) , k = 0 , 1 , . . . model the contribution of presynaptic spikes, which did not trigger the ( k +1) st postsynaptic spike, to the weight change. They are modelled to have i.i.d. components Z 1 ( k ) , . . . , Z d ( k ) , which are supported in [ -( Q -1) , ( Q -1)] , for some Q > 1 , and centred such that E [ Z ( k )] = 0 .

In the remainder of the paper, we analyse the long-run behaviour of p ( k ) as k → ∞ under the learning rule Eq. (3). We say that the output neuron aligns with the j th input neuron if p j ( k ) → 1 as k →∞ . Since the input intensities λ 1 , . . . , λ d > 0 are fixed throughout the dynamic, this condition is equivalent to w j ( k ) / ∑ d i =1 w i ( k ) → 1 as k →∞ . Figure 1 visualises the learning rule Eq. (2).

2 Representation as noisy gradient descent

We continue by relating the learning rule Eq. (1) to noisy gradient descent. For notational simplicity, define Y ( k ) := B ( k ) + Z ( k ) for k = 0 , 1 , . . . . Combining the weight updates Eq. (1) with the formula for the probabilities p from Eq. (2), we find

The normalisation in the denominator and the multiplicative nature of the update ensures that the dynamic of p ( k ) is restricted to the probability simplex. By a Taylor expansion around α = 0 , we find

Since

the random vectors

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p(0)=(0.25,0.4,0.35)

p(0) = (0.4,0.27,0.33)

SS07

SS07

p(0) =(0.22,0.38,0.4)

-0.08

-0.08

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are centred. The distribution of ξ ( k ) depends on w ( k ) and p ( k ) . Up to O ( α 2 ) -terms, we can write the learning rule Eq. (3) as a noisy gradient descent scheme

for the loss function

with gradient

Dropping O ( α 2 ) terms is only done for illustrative purposes. Our main result (Theorem 2.2) applies to the original learning rule Eq. (3). The subsequent lemma summarises the key properties of the loss function L from Eq. (6). For d = 3 , Figure 2 visualises the loss function L and the learning dynamics Eq. (3).

Lemma 2.1. All critical points of the loss function Eq. (6) can be written as p ∗ = 1 | S | ∑ j ∈ S e j for some S ⊆ [ d ] . Every critical point with | S | ≥ 2 is a saddle point. The local minima of the loss function L from Eq. (6) are the standard basis vectors { e 1 , . . . , e d } . Furthermore, every local minimum of L is also a global minimum.

2.1 Linear convergence of the learning rule

We state the convergence guarantee for the learning rule Eq. (3). Renaming the indices, we can assume that p 1 (0) is the largest initial probability. Provided that p 1 (0) is strictly larger than each other component of p (0) , the following theorem shows linear convergence of the first component to 1 on an event Θ in expectation. The probability of Θ can be chosen arbitrarily close to 1 by reducing the learning rate α .

Theorem 2.2. Given ε ∈ (0 , 1) , assume

Then there exists an event Θ with probability ≥ 1 -ε/ 2 such that

Consequently, given δ > 0 ,

Remark 2.3 . If all weights are equal at the starting point of the learning algorithm, the assumption p 1 (0) -max i =2 ,...,d p i (0) > 0 , is equivalent to requiring λ 1 (0) -max i =2 ,...,d λ i (0) > 0 . In this case, the convergence of p ( k ) to e 1 corresponds to the network performing a winner-take-all mechanism [12, 49, 31, 40, 25]. The competitive selection of input neurons in Hebbian STDP has been observed by [39, 38, 15], among others. Our results extend existing findings by going beyond ensemble averages and also provide a rate of convergence for the random dynamics.

The convergence on an event of high probability is in line with other recent results on noisy/stochastic gradient descent for non-convex loss functions, see e.g. [27, 46] or [11, Theorem 2.5]. Contrary to these results, we can choose a constant learning rate and obtain linear convergence. To illustrate the reason for this, we give a brief overview of the proof of Theorem 2.2. The full proof can be found in Section A.1.

We restrict the analysis to the event Θ on which

holds for all iterates k . On this event, the derivative of the first component can be bounded from below by

As described in Eq. (3), we apply a Taylor approximation to the original dynamics. We bound the error term for the i th component p i ( k +1) by the order α 2 p i ( k )(1 -p i ( k )) . The approximation error is dominated by the gradient update on Θ , if the learning rate is small enough (see Eq. (8).

Similarly as in Eq. (5), we restate the learning increments of the dynamics as the sum of the true gradient and a centred noise vector ξ ( k ) . By Eq. (8), this decomposition yields linear convergence of p 1 ( k ) → 1 on Θ .

To find a lower bound for the probability of the chosen event Θ , we employ a similar strategy as Mertikopoulos et al. [27]. Through the representation Eq. (5), we can show that Θ occurs, as soon as M ( k ) := α ∑ k i =1 ξ ( i ) is uniformly bounded by some sufficiently small constant. As ( M ( k )) k ∈ N is a martingale, the probability of the latter event can be controlled through Doob's submartingale inequality (see Eq. (27)).

To apply Doob's submartingale inequality, we bound the second moment of M ( k ) . Since the variance of the components of ξ ( k ) is also dominated by 1 -p 1 ( k ) , we achieve a bound of the order α 2 ∑ ∞ i =1 (1 -p 1 ( i )) /x31 Θ . This series is summable as we have linear convergence to 0 , which allows us to choose a constant learning rate α .

2.2 Associated gradient flow

In this subsection, we consider the associated gradient flow of probabilities p ( t ) as a vector-valued function, which solves the ODE

and is initialized by the probability vector p (0) . By definition, d d t ∑ d i =1 p i ( t ) = 0 , such that ∑ d i =1 p i ( t ) = ∑ d i =1 p i (0) = 1 . Since the updating rule is multiplicative, the gradient flow produces for all t ≥ 0 a probability vector. The gradient flow Eq. (9) also occurs as a specific replicator equation in evolutionary game theory, see Hofbauer and Sigmund [17, Chapter 7]. Elementary properties and an explicit solution for the gradient flow with d = 2 are derived in Section 2.2. Although the loss function L ( p ) in Eq. (6) does not satisfy a global Polyak-Łojasiewicz condition, in particular it is not globally convex, we can deduce the following convergence for the ODE Eq. (9).

Theorem 2.4. Assume for some ∆ > 0 . Then

that is linear convergence of p ( t ) → e 1 as t →∞ .

2.3 Biological plausibility of the proposed learning rule

We study a biological neural network consisting of d input (or presynaptic) neurons, which are connected to one output (or postsynaptic) neuron. For the subsequent argument, we assume that the spike times of the d input neurons are given by the corresponding jump times of d independent Poisson processes ( X (1) t ) t ≥ 0 , . . . , ( X ( d ) t ) t ≥ 0 with respective intensities λ 1 , . . . , λ d . All neurons are excitatory and each connection between an input neuron j ∈ [ d ] and the output neuron has a time varying and nonnegative synaptic strength parameter, which we denote by w j ( t ) ≥ 0 .

An idealized model is that a spike of the j th input neuron at time τ causes an exponentially decaying contribution to the postsynaptic membrane potential of the form t ↦→ w j ( τ ) Ce -c ( t -τ ) /x31 t ≥ τ . We set the parameters c, C > 0 to one, as this can always be achieved by a time change t ↦→ tc and a change of units of the voltage in the membrane potential.

If T j denotes the spike times of neuron j ∈ [ d ] , the postsynaptic membrane potential ( Y t ) t ≥ 0 is given by Y 0 = 0

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and Y t = ∑ d j =1 ∑ τ ∈T j ,τ ≤ t w j ( τ ) e -( t -τ ) for all t ≥ 0 until Y t ≥ S , where S > 0 is a given threshold value. Once the threshold S is surpassed, the postsynaptic neuron emits a spike and its membrane potential is reset to its rest value, which we assume to be 0 . Afterwards, the incoming spikes will contribute to rebuilding the postsynaptic membrane potential. If t 0 := 0 < t 1 < t 2 < . . . denote the postsynaptic spike times, the membrane potential at arbitrary time is therefore given by

We consider the following pair-based spike-timing-dependent plasticity (STDP) rule ([14, Section 19.2.2]): A spike of the j th presynaptic neuron at time τ causes the weight parameter function t ↦→ w j ( t ) to decrease at τ by αe -( τ -t -) , where t -is the last postsynaptic spike time before τ and to increase at any postsynaptic spike time t k by α ∑ τ ∈T j ∩ ( t k , t k +1 ] e -( τ -t k ) , with α > 0 the learning rate. As common in the literature, spike times that occurred before t k only have a minor influence and are neglected in the updating of the weights after t k . The term ∑ τ ∈T j ∩ ( t k , t k +1 ] e -( τ -t k ) is then the trace ([14, Equation (19.12)]) of the j th presynaptic neuron at time t k +1 .

For mathematical convenience, we will assume that all weight-updates in ( t k , t k +1 ] are delayed to the postsynaptic spike times t k in the sense that the learning rule becomes

The postsynaptic spike times t k are the moments at which the postsynaptic membrane potential Y t reaches the threshold S . They depend on the presynaptic spike times, however, the exact dependence is hard to characterise in the assumed model. For mathematical tractability, we will instead work with an adjusted rule to select the postsynaptic spike times t 1 , t 2 , . . . Since Y t only increases at the presynaptic spike times, t k +1 has to happen at a presynaptic spike time. Denote by τ j 1 , τ j 2 , . . . the spike times of the j -th presynaptic neuron after the previous postsynaptic spike time t k in increasing order. The distribution of t k +1 | t k is completely determined by the probabilities

P ( t k +1 = τ jℓ ) = P ( t k +1 = τ jℓ ∣ ∣ t k +1 ∈ ( τ jm ) m ≥ 1 ) P ( t k +1 ∈ ( τ jm ) m ≥ 1 ) , j = 1 , . . . , d, ℓ = 1 , . . . Based on probabilistic arguments related to the underlying Poisson processes that we outline in Section A.3, we replace the probabilities P ( t k +1 ∈ ( τ jm ) m ≥ 1 ) by the probabilities

Working with Eq. (12) instead of P ( t k +1 ∈ ( τ jm ) m ≥ 1 ) results in an approximation of the distribution of t k +1 . Lemma A.7 describes a setting, where Eq. (12) is exact. If all weights are much larger than the threshold S , every presynaptic spike causes a postsynaptic spike. The proof of Lemma A.7 can be adapted to this case to show that the probability that the j th neuron emits the first spike is λ j / ∑ d ℓ =1 λ ℓ . Since Hebbian learning is intrinsically unstable, we argue that the proposed approximation describes the dynamic at the beginning of the learning process. This view is corroborated by experimental results, see point (vi) of Morrison et al. [28, Section 2.1].

Compared to the original definition of t k +1 , the proposed sampling scheme has the advantage that the presynaptic spike times, which were not selected as postsynaptic spike time, add centred noise to the updates. More precisely, one can show that by the construction of t k and the properties of the underlying Poisson processes, the conditional distribution τ |{ τ ∈ ( t k , t k +1 ) } is uniformly distributed on ( t k , t k +1 ) . By the symmetry relation e -( b -u ) -e -( u -a ) = -( e -( b -v ) -e -( v -a ) ) ∈ [ -1 , 1] , which holds for all real numbers a ≤ u ≤ b with v = b + a -u ∈ [ a, b ] , this implies that conditionally on τ ∈ ( t k , t k +1 ) , the random variable e -( t k +1 -τ ) -e -( τ -t k ) is centred and supported on [ -1 , 1] . The update rule Eq. (11) then becomes

with centred random variables Z ( τ, j ) satisfying | Z ( τ, j ) | ≤ 1 . Assuming that the postsynaptic firing rate is slow compared to the learning dynamic, we discard the term e t k -t k +1 glyph[lessmuch] 1 . Since j ∗ ( k +1) follows a multinomial distribution with parameters λ j w j ( t k ) / ( ∑ d ℓ =1 λ ℓ w ℓ ( t k )) , the term /x31 { j = j ∗ ( k +1) } corresponds to the j th component of B ( k ) in Eq. (1). This motivates the learning rule Eq. (1). Additional details on the derivation are given in Subsection A.3 of the supplementary material.

3 A mirror descent perspective

In this section, we rewrite the gradient flow Eq. (9) as natural gradient descent on the probability simplex and relate the discrete-time learning rule Eq. (3) for the probabilities p to noisy mirror gradient descent.

Recall from Eq. (5) that the learning rule Eq. (3) can be interpreted as noisy gradient descent with respect to the loss function L from Eq. (6) in the Euclidean geometry. As we consider a flow on probability vectors, a different perspective is to use the natural geometry of the probability simplex. To this end, we consider the interior of the probability simplex M := int( P ) as a Riemannian manifold with tangent space T p M = { x ∈ R : 1 ⊤ x = 0 } for every p ∈ M . A natural metric on M is given by the Fisher information metric / Shahshahani metric [4, 17], which is induced by the metric tensor d p : T p M×T p M→ R , ( u , v ) ↦→ u ⊤ diag( p ) -1 v at p ∈ M . Here, diag( p ) ∈ R d × d is the diagonal matrix with diagonal entries given by p . We refer to Figure 1 of Mertikopoulos and Sandholm [26] for an illustration of unit balls in this metric. The (Riemannian) gradient of the loss function ˜ L ( p ) = -‖ p ‖ 2 / 2 with respect to d p is given by ∇ d p ˜ L ( p ) = diag( p ) ∇ ˜ L ( p ) ∈ T p M , where we denote by ∇ ˜ L the Euclidean gradient of ˜ L . The Riemannian gradient flow is called natural gradient flow in information geometry [3] and Shahshahani gradient flow in evolutionary game theory [18]. When transforming the Euclidean gradient flow for L to a Riemannian gradient flow on the probability simplex, the part + ‖ p ‖ 2 1 is orthogonal to T p M . Consequently, it does not contribute a direction on the probability simplex. Consequently, the Riemannian gradient flow of ˜ L and the Euclidean gradient flow of L coincide.

The mirror descent algorithm [29] prescribes the discrete-time optimisation algorithm where f : M→ R is the function to be minimised and Φ: M×M→ R + is a suitable proximity function. Euclidean gradient descent is recovered by the choice Φ( p , p ( k )) = ‖ p -p ( k ) ‖ 2 . It

is well-known that the natural gradient flow is the continuous-time analogue of the exponentiated gradient descent or entropic mirror descent , where Φ( p , p ( k )) = KL( p ‖ p ( k )) is chosen as the Kullback-Leibler divergence between p and p ( k ) [2, 47, 33]. Consequently, the gradient flow Eq. (9) can also be viewed as continuous-time version of entropic mirror descent with respect to f = ˜ L . This connection transfers to the discrete-time and noisy updating rule Eq. (3). An alternative approach for connecting our proposed discrete-time learning rule Eq. (3) to entropic mirror descent is included in Subsection A.4 of the supplementary material.

4 Multiple weight vectors

The learning rule Eq. (1) aligns the output neuron with the input neuron of the highest intensity, but no information about the remaining input neurons is unveiled. As a proof-of-concept, we generalise the learning algorithm Eq. (1) to estimate the order of the intensities λ 1 , . . . , λ d . To this end, we consider d different output neurons, which are connected to the d input neurons via the weight vectors w 1 , . . . , w d ∈ R d . The weights at time k are combined into the matrix

and the corresponding probabilities p 1 , . . . , p d are combined into the matrix P ( k ) = [ p 1 ( k ) · · · p d ( k )] ∈ R d . By reordering the neurons, we can achieve λ 1 ≤ λ 2 ≤ · · · ≤ λ d . If the intensities are strictly ordered, our goal is the alignment of the j th output neuron with the j th input neuron, which amounts to the convergence of P ( k ) to the identity matrix I ∈ R d × d as time

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increases. If multiple intensities are equal, convergence is up to permutations within the group of equal intensities. We propose Algorithm 1, which constitutes an STDP rule as lines 3 - 4 can be implemented using the spike-trains and the learning rule Eq. (11).

Algorithm 1: Aligning multiple output neurons

Algorithm 1 is inspired by Sanger's rule [35] for learning d principal components in streaming PCA. The first weight vector w 1 ( k ) aligns with e 1 by Theorem 2.2 since its dynamic equals the learning rule Eq. (1). By removing the components of the change ∆ w j ( k ) in the direction of w 1 ( k ) , . . . , w j -1 ( k ) in line 5 of Algorithm 1, the weight vector w j ( k ) is forced to converge to e j , similarly to the Gram-Schmidt algorithm.

Simulations of the corresponding probability matrix P ( k ) with varying learning rates and Z drawn i.i.d. from Unif([ -1 , 1] d ) are included in Figure 5. We choose different learning rates for the d vectors satisfying α 1 > · · · > α d > 0 . This ordering ensures fast convergence of the lower order weight vectors to the correct standard basis vector and counteracts the impact of initial misalignments of the higher order weight vectors. The simulation of Algorithm 1 shown in Figure 5 displays a decrease

P(k)

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[P(k)  /²/2 with learming rates 103[1.0, 0.75, 0.5]

[P(k)  /²/2 with learming rates 10-4[1.0, 0.75, 0.5]

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of the Frobenius error ‖ P ( k ) -I ‖ 2 / 2 over the iteration index k , when averaged (blue line). Nevertheless, we observe that for a single trajectory, the error can plateau around 1 and 2. Given that the probability vectors tend to converge to standard basis vectors { e 1 , . . . , e d } , a non-vanishing error is due to an incorrect ordering or duplicates. Consequently, the error ‖ P ( k ) -I ‖ 2 / 2 corresponds to the number of output neurons aligning with the incorrect input neuron, and plateaus at 1, 2 and 3 can arise. Theorem 2.2 shows that this phenomenon can be mitigated by slower learning rates, which is corroborated by decreasing the base learning rate from 10 -3 to 10 -4 in the simulation. A rigorous mathematical analysis of the Algorithm 1 is challenging due to joint updates in all read-out neurons. In Section A.5 we show that Theorem 2.2 is applicable if the learning is split into disjoint learning periods, and we derive theoretical convergence guarantees.

5 Extensions, discussion and limitations

Time-inhomogeneous intensities. We considered input spike trains generated from Poisson point processes with fixed intensity. It is natural to extend this to time-inhomogeneous intensities. Here we assume that the intensities of the input neurons are constant on the interval ( k, k +1] and are stored in the vector λ ( k ) . The intensities λ ( k ) and weights w ( k ) determine the spike-triggering-probabilities p ( k ) = λ ( k ) glyph[circledot] w ( k ) / λ ( k ) ⊤ w ( k ) = E [ Y ( k )] and the update formula Eq. (2) becomes

˜ with ˜ p ( k ) := λ ( k +1) glyph[circledot] w ( k ) / ( λ ( k +1) ⊤ w ( k )) . A first order Taylor expansion yields

Since E [ Y ( k )] = p ( k ) , this means that p ( k +1) = ˜ p ( k ) glyph[circledot] ( 1 + α ( p ( k ) -˜ p ( k ) ⊤ p ( k ) 1 ) ) + centered noise + O ( α 2 ) . (17) Extending the gradient flow derivation to the time-inhomogeneous case, one can identify the ODE

where the logarithm is taken componentwise, as corresponding deterministic dynamic in continuous time, see Section A.6 for a derivation. The ODE can be interpreted as a replicator equation with (time-varying) fitness d d t log( λ ( t )) + p ( t ) , see e.g. Chapter 7 of [18]. An interesting scenario which lies beyond our mathematical analysis amounts to considering time-dependent mean firing rates that are piecewise constant, corresponding to the successive exposition to different input patterns [9].

Correlated inputs. Correlated inputs facilitate simultaneous spiking of different input spikes. The probability that input i and j spike at the same time is denoted by Γ ij , and naturally Γ ii := 1 , for all i ∈ [ d ] . We introduce the d × d random symmetric matrix C ( k ) = ( C i,j ( k )) i,j ∈ [ d ] with independently sampled entries

C ( k ) describes the simultaneous spiking of the different inputs at the k -th post-synaptic spike, i.e. if C i,j ( k ) is 1 then inputs i and j both spike at the k -th post-synaptic spike if either i or j caused the post-synaptic spike. Compared to the original model, the random vector Z ( k ) remains the same, but the random vector B ( k ) = ( B 1 ( k ) , . . . , B d ( k )) ⊤ that encapsulates which of the presynaptic neurons caused the postsynaptic spike is replaced by S ( k ) = C ( k ) B ( k ) . S ( k ) encodes which of the inputs spike at a post-synaptic spike, and in particular it holds where Γ i, · ( k ) denotes the i -th row of Γ ( k ) . Since the only change in the dynamic of p ( k ) is replacing B ( k ) by S ( k ) in the definition of Y ( k ) , Eq. (4) still holds true, and we obtain

which induces the following gradient flow

This is again a replicator equation with fitness Γp , compare Section 7 of [18]. The associated Shahshahani-loss is given by x ↦→ -1 2 x ⊤ Γx . Thus, in the correlated model, the probabilities p follow a flow restricted to the probability simplex aimed at maximising the quadratic form associated to the matrix Γ . This property is similar to principal component analysis (PCA), as the goal there is to recover the eigenvector corresponding to the largest eigenvalue of the underlying covariance matrix. Thus, the behaviour of the correlated model can be interpreted as a form of PCA restricted to the probability simplex. Theorem A.13 in the supplementary material generalizes Theorem 2.2 to weakly correlated input neurons and is accompanied by simulations in Figure 6.

Small biological neural network. In this paper, we mathematically analyse the convergence behaviour of a small biological neural network with one layer composed of excitatory presynaptic/input neurons and multiple postsynaptic/output neurons. It is natural to generalise the setting to account for inhibitory neurons and more than one layer.

Weight explosion. The learning rule Eq. (1) for the weights w ( k ) causes them to increase without bound as the iteration index k increases. When the weights exceed the spike threshold, the model becomes biologically implausible and the derivation of the probabilities Eq. (12) is no longer valid. This unstable nature is well-known to be intrinsic to Hebbian learning algorithms and is commonly countered by soft or hard bounds, or by including mean-reverting terms to the dynamic Gerstner et al. [14, pages 497-498]. We follow a different route, namely viewing Hebbian learning as a temporal phase of limited length, which is followed by a stabilising homeostatic learning phase. This view is corroborated by experimental results, compare Point (vi) in Morrison et al. [28, Section 2.1].

Beyond Pair-based STDP rules. While pair-based learning rules such as Eq. (1) only account for the relative timing of one pre- and one postsynaptic spike time, also the voltage at the location of the synapse should be taken into account [37]. A natural generalization of our framework would be to extend the results to the model proposed in [9].

Acknowledgments and Disclosure of Funding

All authors acknowledge support from ERC grant A2B (grant agreement no. 101124751). S. G. has been partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) CRC/TRR 388 "Rough Analysis, Stochastic Dynamics and Related Fields Project ID 516748464, Project B07. Parts of the research were carried out while the authors visited the Simons Institute in Berkeley. We thank Xiangyuan Li for pointing out some typos.

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A Technical Appendix

We first study the loss landscape

Lemma 2.1 identifies the stationary points if we view the landscape as a function on R d .

Proof of Lemma 2.1. The formula Eq. (7) for the gradient ∇ L ( p ) shows that the set of critical points is given by

To identify local the extrema, we compute the Hessian matrix

where diag( p ) ∈ R d × d is the diagonal matrix with diagonal entries given by p . Substituting a critical point p ∗ with n ∈ [ d ] non-zero entries yields (up to permutations of rows and columns)

where 1 n × n is the n × n matrix consisting of ones. The corresponding eigenvalues are where E ⊥ n is the orthogonal complement of E n in span( e 1 , . . . , e n ) . Consequently, only those critical points p ∗ ∈ Crit are local minima, which have n = 1 , i.e. p ∗ ∈ { e 1 , . . . , e d } . Since all local minima attain the same loss -1 / 12 and L ( p ) → ∞ as ‖ p ‖ → ∞ , every local minimum is also a global minimum.

Remark A.1 . The eigenvalues of the Hessian of the loss function computed in Eq. (20) also imply that if n ≥ 2 , then p ∗ ∈ Crit is a saddle point in R d . Interestingly, when restricting to directions within the probability simplex, the case n = d is not a saddle point, but a maximum, since the direction ∑ d i =1 e i is orthogonal to P .

A.1 Proofs for Subsection 2.1

In the following we will always assume that

for some ∆ ∈ (0 , 1) . This is a deterministic constraint. The randomness occurs because of the noise in the updates. We assume that all random variables are defined on a filtered probability space (Ω , F , P ) , and denote by F n , n = 0 , 1 , . . . , the natural filtration of ( B ( n ) , Z ( n )) n ∈ N . By a slight abuse of notation we also introduce F -1 = {∅ , Ω } . In particular it then holds that p n is F n -1 -measurable for n = 0 , 1 , . . . . The starting point for the proof of the linear convergence of STDP is given by the following Lemma, which explicitly bounds the error term in the Taylor approximation contained in Eq. (3). Recall that | Y i ( k ) | ≤ Q , for all i ∈ [ d ] , k = 0 , 1 , . . .

Lemma A.2. For i ∈ [ d ] and k = 0 , 1 , . . . define

and assume α < 1 /Q . Then for any i ∈ [ d ] and k = 0 , 1 , . . . , there exists a random variable θ i ( k ) , satisfying such that

Proof. By definition

Thus, a Taylor expansion around x = 0 gives that there exists some γ ∈ (0 , x ) , such that

Hence we obtain that for some γ ∈ (0 , α ) ,

Using that | Y i ( k ) | ≤ Q almost surely for all i ∈ [ d ] , k = 0 , 1 , . . . , the absolute value of the error term can be bounded as follows

̸

Since p i ( k ) is F k -1 -measurable for any i ∈ [ d ] and E [ Y i ( k ) | F k -1 ] = p i ( k ) , we also obtain

which concludes the proof.

Now, for a, b ∈ [ -Q,Q ] the first two derivatives of the function are given by

For ∆ given in Eq. (21), we define a sequence of benign events and due to the assumption Eq. (21) we set Ω(0) = Ω . On the above events, the gradient is bounded away from zero and

Proposition A.3. If

then, on the event Ω( k ) ,

Proof. By definition, we have on the event Ω( k ) ,

The constraint imposed on the learning rate implies that α < 1 /Q and Lemma A.2 becomes applicable. Now, combining the previous inequality with the assumption on α and applying Lemma A.2 with i = 1 , as well as Eq. (23), gives, on the event Ω( k ) ,

This concludes the proof.

Having understood the dynamics of p on the favourable event Ω( k ) , we aim for a lower bound for its probability. A key step is the following Lemma, which states that Ω( k ) is fulfilled as soon as

with ξ j ( ℓ ) defined in Eq. (22), exhibit a uniform concentration behaviour.

Lemma A.4. Define the sets

Then if

the following set inclusion holds for any k = 0 , 1 , . . .

Using these properties, we can prove a recursive upper bound for 1 -p 1 ( k ) .

Proof. Let u ∈ { 2 , . . . , d } be arbitrary. It follows by Lemma A.2, that on Ω( k ) the bound holds. Consequently, on Ω( k ) , we have

We have p u ( k ) ≤ 1 -p 1 ( k ) and thus, on Ω( k ) ,

invoking the constraint on the learning rate in the last step. From the assumptions on α we deduce that on the event Ω( k ) ,

where we applied Eq. (24) in the third to last inequality. Because of Ω( k ) ⊆ Ω( k -1) it then follows,

This gives

We want to prove by induction that E ( k ) ⊆ Ω( k +1) for all k = 0 , 1 , . . . . For k = 0 , this directly follows from Eq. (26), since Ω(0) = Ω due to assumption Eq. (21). Assume the assertion holds for some k = 0 , 1 , . . . . Hence, it holds E ( k +1) ⊆ E ( k ) ⊆ Ω( k +1) , such that for any u ∈ { 2 , . . . , d } it holds on E ( k +1) by Eq. (26)

which proves the assertion.

Having assembled the previous results, we are able to prove the linear convergence of STDP stated in Theorem 2.2. As Proposition A.3 already suggests the desired behaviour of p on Ω( k ) , the main

part of the proof is to show that Ω( k ) is satisfied with large probability. For that we deploy Doob's submartingale inequality, which states that for a martingale ( X n ) n ∈ N , any p ≥ 1 , and any u > 0 ,

This will be applied to derive lower bounds for the event E ( k ) defined in Lemma A.4, which are also lower bounds for the probability of Ω( k +1) by the same Lemma. For the reader's convenience we restate Theorem 2.2 before giving its proof.

Theorem 2.2. Given ε ∈ (0 , 1) , assume

Then there exists an event Θ with probability ≥ 1 -ε/ 2 such that

Consequently, given δ > 0 , it holds

Proof of Theorem 2.2. The recursive definition ensures that p ( k ) is F k -1 -measurable. Thus, also Ω( k ) ∈ F k -1 for any k = 0 , 1 , . . . . One can check that ( M i ( k )) k =0 , 1 ,... , defined in Eq. (25), forms a martingale for each i ∈ [ d ] . This allows us to apply Doob's submartingale inequality. To apply it with p = 2 , we deduce the following bound on the second moment,

E [ M 1 ( k ) 2 ]

̸

̸

where we used that E [ ξ 1 ( k 2 ) | F k 1 ] = 0 , for any k 2 > k 1 = 0 , 1 , . . . , together with | Y 1 ( k ) | ≤ Q , the inequality ( a + b ) 2 ≤ 2( a 2 + b 2 ) for a, b ∈ R and 1 -p 1 ℓ ≤ 1 . Arguing similarly we obtain for u ∈ { 2 , . . . , d } ,

Hence, applying a union bound, Doob's submartingale inequality Eq. (27) with p = 2 gives for any k = 0 , 1 , . . .

Proposition A.3 gives for any k = 0 , 1 , . . . the bound

E [(1 -p 1 ( k +1)) /x31 Ω( k +1) ] ≤ E [(1 -p 1 ( k +1)) /x31 Ω( k ) ] ≤ E [( 1 -α ∆ 4 d ( 1 + ∆ 2 ( d -1) )) ( 1 -p 1 ( k ) ) /x31 Ω( k ) + αξ 1 ( k ) /x31 Ω( k ) ] = ( 1 -α ∆ 4 d ( 1 + ∆ 2 ( d -1) )) E [ ( 1 -p 1 ( k ) ) /x31 Ω( k ) ] , which implies

We set

The continuity of probability measures and Lemma A.4 then imply

where we used that we can assume d ≥ 2 without loss of generality. Additionally, Eq. (28) and the elementary inequality 1 -x ≤ exp( -x ) , which is valid for any real number x, give

When d = 1 , the right hand side of this inequality is 0 . For d ≥ 2 , we can also use the bound d -1 ≥ d/ 2 . Together with

this concludes the proof of the first statement. For the proof of the second statement, we apply Markov's inequality to obtain

Hence, if then,

Lemma A.5. Given an initialization of the weights w (0) = ( w 1 (0) , . . . , w d (0)) ⊤ consider two d -dimensional intensity vectors λ = ( λ 1 , . . . , λ d ) ⊤ , ˜ λ = ( ˜ λ 1 , . . . , ˜ λ d ) ⊤ with positive entries. Assume that λ 1 w 1 (0) > max i =2 ,...,d λ i w i (0) and ˜ λ d w d (0) > max i =1 ,...,d -1 ˜ λ i w i (0) . Assume we run the learning rule Eq. (3) with intensity λ until time K ∗ and then, for k > K ∗ , change the intensity to ˜ λ and run the learning dynamic until time k →∞ . If K ∗ is small, in particular, if K ∗ = 0 , the above convergence result can be applied to show that the dynamic converges to the corner e d . However, for any ε ∈ (0 , 1) and all sufficiently large K ∗ (depending on ε ), the dynamics will converge to e 1 with probability 1 -2 ε.

This result shows that the dynamic can be primed at the beginning to end up in one regime. Despite the noise and the infinite amount of data, the dynamic is unable to escape this domain of attraction. From the proof, one can derive quantitative bounds for K ∗ .

Proof. Given ε ∈ (0 , 1) , choose δ ∈ (0 , 1) such that

̸

Let ∆ = p 1 (0) -max i =1 p i (0) . Given ε, δ, ∆ choose

̸

On the event ‖ p ( K ∗ ) -e 1 ‖ 1 < δ, we have

which can be combined into

̸

Using the inequality Eq. (29), we obtain

̸

This means that restarting the learning rule Eq. (3) at time K ∗ with intensities ˜ λ 1 , . . . , ˜ λ d and weights w 1 ( K ∗ ) , . . . , w d ( K ∗ ) , shows that p 1 ( K ∗ ) > max i =2 p i ( K ∗ ) . Applying Theorem 2.2 again shows convergence to e 1 with probability 1 -2 ε.

A.2 Proofs for Subsection 2.2

This section contains additional material on the gradient flow Eq. (9), as well as the proofs for Lemma A.6 and Theorem 2.4.

For d = 2 , the gradient flow admits an explicit solution. In this case, p ( t ) = ( p 1 ( t ) , p 2 ( t )) ⊤ . If p (0) = (1 / 2 , 1 / 2) ⊤ , then this is a stationary solution and p ( t ) = p (0) = (1 / 2 , 1 / 2) ⊤ for all t ≥ 0 . If p 1 (0) > 1 / 2 , then,

If p 1 (0) < 1 / 2 , then p 2 ( t ) = 1 -p 1 ( t ) > 1 / 2 follows the dynamic in Eq. (30). This formula immediately implies that p 1 ( t ) converges exponentially fast to 1 .

Proof of Formula Eq. (30). Throughout the proof we set p ( t ) := p 1 ( t ) and do not use the previous notation p 1 ( t ) , p 2 ( t ) for the first and second probability. For d = 2 , the gradient flow ODE Eq. (9) becomes

Rewriting this in the variable u ( t ) = 1 -2 p ( t ) gives the dynamic,

̸

̸

This is solved by u ( t ) = -1 / √ Ce -t +1 since

Thus, p ( t ) = 1 2 (1 -u ( t )) solves Eq. (31). Finally, C is determined by the initial condition p (0) = 1 2 (1 -1 / √ C +1) .

The following lemma summarises different properties of the gradient flow Eq. (9). In its statement, differentiable on [0 , 1] means differentiable on (0 , 1) and continuous on [0 , 1] .

Lemma A.6. The gradient flow Eq. (9) exhibits the following properties.

(a) If ϕ : [0 , 1] → R is a convex and differentiable function, then t ↦→ ∑ d i =1 ϕ ( p i ( t )) is monotone increasing.

glyph[negationslash]

(b) Let i, j ∈ [ d ] with i = j . If p i (0) > p j (0) , respectively p i (0) = p j (0) , then p i ( t ) > p j ( t ) , respectively p i ( t ) = p j ( t ) , for all t ≥ 0 . Moreover, if

Lemma A.6 implies that the q -norm t ↦→| p ( t ) | q is monotonically increasing whenever 1 ≤ q < ∞ . The result also implies that if instead ϕ is concave and differentiable, then t ↦→ ∑ d i =1 ϕ ( p i ( t )) is monotonically decreasing.

Proof of Lemma A.6.

(a) Since ϕ is convex, ϕ ′ is monotonically increasing. Thus, for a probability vector q = ( q 1 , . . . , q d ) , we have

(If ϕ is strictly convex, then strict equality holds if and only if q is one of the stationary points described above.) Using this and the gradient flow formula

proving the result.

(b) By definition it holds for i, j ∈ [ d ]

From this we deduce that

then

which concludes the proof of the first statement since the exponential function is strictly positive. To prove the second statement, we have

and similarly, for any i ∈ { 2 , . . . , d } ,

Hence, whenever t ∈ [0 , ∆ / 2] , we find

which also implies

for all t ∈ [0 , ∆ / 2] . Therefore for any t ∈ [0 , ∆ / 2] , i ∈ [ d ] it holds

which implies for any i ∈ { 2 , . . . , d } and t ∈ [0 , ∆ / 2] ,

Applying this argument iteratively concludes the proof.

For the reader's convenience we restate Theorem 2.4 before giving its proof.

Theorem 2.4. Assume for some ∆ > 0 . Then

that is linear convergence of p ( t ) → e 1 as t →∞ .

Proof of Theorem 2.4. Arguing as in Eq. (24) and Eq. (23), Lemma A.6 Eq. (b) implies

where

Grönwall's inequality entails which gives

A.3 Proofs for Subsection 2.3

In this subsection, we heuristically derive the expression for the probabilities

in Eq. (12). To this end, we assume that the weights are small compared to the threshold S , that the weights are only updated at the postsynaptic spike times, and that ∑ d ℓ =1 λ ℓ w ℓ ( t k ) glyph[greatermuch] S . For convenience, we write w ℓ for w ℓ ( t k ) and all ℓ ∈ [ d ] . The constraint ∑ d ℓ =1 λ ℓ w ℓ glyph[greatermuch] S guarantees that after the postsynaptic spike time t k , the membrane potential Y t will again reach S and thus emit another spike at time t k +1 .

Taking the expectation of the membrane potential Y t = ∑ d j =1 ∑ τ ∈T j ∩ ( t k ,t ] w j e -( t -τ ) with respect to all except the j th spike-train, gives

̸

for all t k ≤ t < t k +1 .

̸

Introduce t ∗ := inf { t ≥ t k : Z t ≥ S -w j } and write t + for the first time after t ∗ where

̸

reaches the threshold S . If there are sufficiently many neurons, the probability that the j th presynaptic neuron spikes at time t ∗ is small and will be neglected. We have Z t ∗ = S -w j such that V t + = w j . Approximating 1 -e -( t + -t ∗ ) ≈ t + -t ∗ gives

̸

The j th presynaptic neuron causes the next postsynaptic spike if and only if it spikes in the interval ( t ∗ , t + ) . The spike times of the j th presynaptic neuron are generated from a Poisson process with intensity λ j . Thus, if U ∼ Poisson( λ j ( t + -t ∗ )) , the probability that the j th presynaptic neuron spikes in ( t ∗ , t + ) is given by glyph[negationslash]

We can moreover approximate the denominator on the right hand side by the full sum ∑ d ℓ =1 w ℓ λ ℓ . Since the probabilities add up to one, we must have e t ∗ -t k ≈ 1 . This shows that the probability of the j th presynaptic neuron triggering the first postsynaptic spike after t k is approximately given by Eq. (33).

̸

LemmaA.7. Consider the setting outlined in Subsection 2.3. If at some time point t > 0 , all weights are the same, then the probability that the j th neuron triggers the next postsynaptic spike after t is given by

Proof. Since all weights are the same, we can denote their value by w . The j th neuron causes a postsynaptic spike if and only if it is the first one to spike after the postsynaptic membrane potential Y t has reached a level ≥ S -w . As t ∗ = inf { t ≥ t : Y t ≥ S -w } is a jump time and a stopping

time, we can restart the process at t ∗ . As the increments of Poisson processes are independent, and the time between the jumps is exponentially distributed with parameters λ j , the probability that the j th neuron causes the next presynaptic spike is given by

where ( X i ) i ∈ [ d ] are independent random variables satisfying X i ∼ Exp( λ i ) . If U ∼ Exp( λ ) and V ∼ Exp( λ ′ ) are independent, then, U ∧ V ∼ Exp( λ + λ ′ ) and P ( U ≤ V ) = λ/ ( λ + λ ′ ) . Thus, min i = j X i ∼ Exp( ∑ i = j λ i ) , and

̸

̸

The above lemma and the previous discussion give a motivation for the form of the probabilities in settings, where the weights are small compared to the threshold S or equal. We now give another heuristic, motivating our modelling choice. Let Y be the postsynaptic membrane potential. Then, in expectation Y grows linearly with slope λ ⊤ w . Furthermore, input i causes a postsynaptic spike if, and only if, it spikes at a time at which Y ≥ S -w i , where S > 0 is the threshold level. As Y 's growth is approximately linear, the amount of time in which Y ≥ S -w i , holds is approximately equal to w i / λ ⊤ w . Now the probability that input i jumps in an interval of length w i / λ ⊤ w is given by 1 -exp( -λ i w i / λ ⊤ w ) ≈ λ i w i / λ ⊤ w , which is exactly our modelling choice in the independent setting.

A.4 On the connection to entropic mirror descent

An alternative approach to connecting our proposed learning rule Eq. (3) for the probabilities p and the entropic mirror descent in discrete-time is as follows. The entropic mirror descent step Eq. (14) with Kullback-Leibler divergence and potential f can be solved explicitly and yields

see Section 5 of Beck and Teboulle [5] for details. With f ( p ) = ˜ L ( p ) = ‖ p ‖ 2 / 2 and the first order approximation exp( x ) ≈ 1 + x for small x we deduce for any i = 1 , . . . , d and k = 0 , 1 , . . . . As our proposed learning rule Eq. (3) is a noisy version of the last line, it is naturally connected to noisy entropic gradient descent.

A.5 Theoretical results for the alignment of multiple read-out neurons

For a weight vector w ∈ R d with nonnegative entries let i ∗ := argmax i =1 ,...,d w ⊤ e i = argmax i =1 ,...,d w i and define the cosine-projection

Assume that λ 1 > · · · > λ d and consider first learning the first weight vector w 1 using the learning rule in Eq. (1) while the remaining weight vectors w 2 , . . . , w d are fixed. Theorem 2.2 implies that after K iterations, we have w 1 ≈ e 1 . By projecting onto w ∗ 1 := P w 1 ( K ) and setting w 2 (0) ← w 2 (0) -w ∗ 1 w 2 (0) ⊤ w ∗ 1 / ‖ w ∗ 1 ‖ , we ensure that [ p 2 (0)] 2 = max i =1 ,...,d [ p 2 (0)] i and Theorem 2.2 applies and yields p 2 ( K ) ≈ e 2 . Proceeding successively, we arrive at Algorithm 2.

̸

Theorem A.8. Consider Algorithm 2. Assume that λ 1 > · · · > λ d and the minimal gap ∆ = min i =1 ,...,d -1 ([ p i (0)] i -max j>i [ p i (0)] j ) > 0 is positive. Then we have

glyph[negationslash]

as K →∞ . More precisely, let δ < κ/ (1 + κ ) for κ = min i =1 ,...,d λ i / max i =1 ,...,d λ i and ε > 0 . Then

Algorithm 2: Sequential alignment of multiple output neurons

Before proving Theorem A.8, we start with an auxiliary result on the order of the weights when the probability vector is close to a standard unit vector.

Lemma A.9. Assume that 0 < λ min = min i =1 ,...,d λ i ≤ λ max = max i =1 ,...,d λ i < ∞ and let κ = λ min /λ max . Consider a weight vector w ∈ R d with corresponding probability vector p = w glyph[circledot] λ / w ⊤ λ and let 0 < δ < 1 . Then the condition 1 -p 1 < δ implies that max i =2 ,...,d w i ≤ w 1 κ -1 δ/ (1 -δ ) .

Proof. Since 1 -p 1 ≤ δ we know that ∑ d i =2 λ i w i ≤ δ 1 -δ λ 1 w 1 . By bounding the λ i from above and below we find such that

Proof of Theorem A.8. By Theorem 2.2 we know that P ( ‖ p 1 ( K ) -e 1 ‖ 1 ≤ δ ) ≥ 1 -ε for our choice of K . Note that the projection picks out the direction argmax i =1 ,...,d w 1 ( K ) ⊤ e i = argmax i =1 ,...,d [ w 1 ( K )] i . Consequently, Lemma A.9 and δ < κ/ (1 -κ ) imply that w 1 ( K ) projects in the direction of e 1 with probability at least 1 -ε . We find that P ( p ∗ 1 = e 1 ) ≤ 1 -ε . By the adjustment of the weight vector w 2 (0) we remove its first component, such that [ p 2 (0)] 1 = 0 and [ p 2 (0)] 2 = max j =1 ,...,d [ p 2 (0)] i . By the assumption on [ p 2 (0)] 2 -max j =3 ,...,d [ p 2 (0)] j ≥ ∆ > 0 we find that P ( ‖ p 2 ( K ) -e 2 ‖ 1 ≤ δ ) ≥ 1 -ε . By iteration and independence of the training windows we conclude the proof.

glyph[negationslash]

A.6 Additional material for the extension to time-inhomogeneous intensities

Derivation of Eq. (18): Eq. (17) yields the deterministic update scheme

˜ ˜ Let T > 0 be the time horizon and assume that the time-inhomogeneous intensities λ α change on the same scale as the weights and are given by λ α ( t ) = λ ( tα ) , t ∈ [0 , T/α ] , where λ ∈ C 2 b ([0 , T ]) is a universal, twice differentiable function with bounded derivatives and λ ( t ) ≥ λ min > 0 , t ∈ [0 , T ] , componentwise. For vectors w , λ , λ ′ that are of the same length, and ∆ := λ ′ -λ , we have

Applying this with λ ′ = λ α ( k +1) , λ = λ α ( k ) , w = w ( k ) , yields

Our assumptions on λ α imply that ˜ p ( k ) -p ( k ) ≲ ‖ ∆ λ α ( k ) ‖ ≲ α for all k = 0 , 1 , . . . , glyph[floorleft] T/α glyph[floorright] . This gives ˜ p ( k ) glyph[circledot] p ( k ) -˜ p ( k ) ˜ p ( k ) ⊤ p ( k ) = p ( k ) glyph[circledot] p ( k ) -p ( k ) p ( k ) ⊤ p ( k ) + O ( α ) . In combination with Eq. (35) we find

Sending α → 0 and using that ∆ λ α ( t/α ) /α → d d t λ ( t ) as α → 0 we recognize Eq. (36) as an Euler-type scheme for the ODE

where the logarithm is taken componentwise.

A.7 Analysis of the correlated model

glyph[negationslash]

In the following we always assume that the off-diagonal entries of Γ are strictly smaller than 1 , that is, Γ i,j < 1 for all i = j = 1 , . . . , d. This assumption is reasonable since a perfect correlation between input i and input j corresponds to one single input neuron with Poisson process intensity λ i + λ j . Under this assumption it is easy to see that the quadratic form associated to Γ is maximised on the probability simplex by the basis vectors e 1 , . . . , e d . Thus, the natural question to investigate is the same as in the original model: To which basis vector does the model converge? In the following section we investigate this question and show results for the weakly dependent case.

p(0) = (0.25,0.4,0.35)

0.25

0.25

0.25

p(0) = (0.4,0.27,0.33)

p(0) = (0.22,0.38,0.4)

SSO"

SSOT

0.50

0.50

0.50

0.23

0.23

0.23

p(0) = (0.25,0.4,0.35)

p(0) = (0.4,0.27,0.33)

p(0) = (0.22,0.38,0.4)

SSO"

LoSS

0.50

0.50

0.50

0.19

0.19

0.19

p(0) =(0.25,0.4,0.35)

p(0) = (0.4,0.27,0.33)

p(0) = (0.22,0.38,0.4)

0.50

0.50

0.50

Contour plot of the Shahshahani loss function L ( p ) = -1 2 p ⊤ Γp on the probability simplex P for d = 3 with different overlays. Left: Three sample trajectories of Eq. (3) with different initial configurations p (0) . Middle: Stream plot of the gradient field given by Eq. (7). Right: 100 sample trajectories of Eq. (3) with p (0) = (0 . 3 , 0 . 3 , 0 . 4) ⊤ . All trajectories are simulated with 2000 iteration steps, learning rate α = 0 . 01 and Z ( k ) ∼ Unif([ -1 , 1] d ) .

The analysis of the correlated version of the model follows the same steps as in the independent case. For this we assume that all random variables are defined on a filtered probability space (Ω , F , P ) and denote by F n , n = 1 , 2 , . . . the natural filtration of ( B ( n ) , Z ( n ) , C ( n )) n ∈ N , and set F -1 = {∅ , Ω } . The proof follows exactly the same steps as the proof of Theorem 2.2. We start with the following result, which bounds the error of the Taylor approximation of the random dynamics.

Lemma A.10. For i ∈ [ d ] and k = 0 , 1 , . . . define

and assume α < 1 /Q . Then for any i ∈ [ d ] and k = 0 , 1 , . . . , there exists a random variable θ i ( k ) , satisfying such that

Proof. As the dynamic of p still follows Eq. (4) as in the uncorrelated model, we can argue as in the proof of Lemma A.2 to obtain that for some γ ∈ (0 , 1)

Then, since | Y i ( k ) | ≤ Q we obtain the same bound on the error term as in the proof of Lemma A.2

Since the p i ( k ) are F k -1 -measurable and C ( k ) is independent of F k -1 , we finally obtain E [ Y i ( k ) |F k -1 ] = Γ i, · p ( k ) .

In the following we always assume for ∆ p , ∆ Γ > 0

These assumptions are for example fulfilled in the following case

since we can choose ∆ p = 0 . 7 , ∆ Γ = 0 . 64 , ν = 0 . 1 , and hence c ⋆ = 8 ∗ 10 -4 . In the independent case, which corresponds to ν = 0 , the assumptions given in Eq. (38) reduce to the original assumption given in Eq. (21). For k = 1 , 2 , . . . , we define a sequence of benign events

Proposition A.11. If

then, on the event Ω( k ) ,

Proof. By definition, we have on the event Ω( k ) ,

Due to assumption Eq. (38), Ω(0) = Ω . Additionally, since the assumptions in Eq. (38) imply the gradient to be bounded away from 0 , we can prove the following recursive upper bound for 1 -p 1 ( k ) .

̸

The assumption on α implies α < 1 /Q and thus Lemma A.10 becomes applicable. Now applying Lemma A.10 with i = 1 , gives on the event Ω( k ) ,

This concludes the proof.

As in the uncorrelated setting our goal is now to derive a lower bound for the probability of the favourable event Ω( k ) . For this we again follow the same strategy and rely on uniform concentration inequalities for martingales. In order apply those, we require the following lemma, which states that Ω( k ) is fulfilled as soon as

with ξ j ( ℓ ) defined in Eq. (22), exhibits a uniform concentration behaviour. In the following we denote by ‖ Γ ‖ ∞ the row-sum norm of Γ , i.e. ‖ Γ ‖ ∞ = max i ∈ [ d ] ∑ d j =1 Γ i,j . Lemma A.12. Define the sets

Then if

the following set inclusion holds for any k = 0 , 1 , . . .

Proof. Let u ∈ { 2 , . . . , d } be arbitrary. By Lemma A.10, on the event Ω( k ) ,

Since p u ( k ) ≤ 1 -p 1 ( k ) we also obtain on Ω( k ) ,

The assumptions on α then imply that on the event Ω( k ) ,

Because of Ω( k ) ⊆ Ω( k -1) it then follows,

This gives

Now again let u ∈ { 2 , . . . , d } be given. Then it holds, on Ω( k )

̸

Hence, arguing as in the derivation of Eq. (41) gives

With the above results we can now begin proving that E ( k ) ⊆ Ω( k +1) for all k = 0 , 1 , . . . . We do this by induction. For k = 0 , this directly follows from Eq. (41) and Eq. (42), since Ω(0) = Ω

by assumption. Now, assume the assertion holds for some k = 0 , 1 , . . . . This implies E ( k +1) ⊆ E ( k ) ⊆ Ω( k +1) , such that for any u ∈ { 2 , . . . , d } it holds on E ( k +1) by Eq. (41)

and additionally by Eq. (42) it holds on E ( k +1)

which proves the assertion.

With the above results we are now able to prove and state the main theorem for the correlated case.

Theorem A.13. Given ε ∈ (0 , 1) , assume and

̸

Then there exists an event Θ with probability ≥ 1 -ε/ 2 such that

Consequently, given δ > 0 , it holds

Before giving the proof of the above theorem, we want to remark that Theorem A.13 exactly recovers the result of Theorem 2.2 in the independent case. Indeed, in the independent case Γ is equal to the identity matrix and thus ∆ p = ∆ Γ , ν = 0 and c ⋆ = ∆ 2 p / 4 hold true, which gives the result of Theorem 2.2.

Proof of Theorem A.13. As in the uncorrelated case, the recursive definition ensures that p ( k ) is F k -1 -measurable. Thus, also Ω( k ) ∈ F k -1 for any k = 0 , 1 , . . . . Then ( M i ( k )) k =0 , 1 ,... , defined in Eq. (40), is a martingale for each i ∈ [ d ] . This allows us to apply Doob's submartingale inequality. For this, we deduce the following bound on the second moment,

where we argued similarly as for the independent case. Furthermore, we obtain for u ∈ { 2 , . . . , d } ,

Hence, applying a union bound, Doob's submartingale inequality Eq. (27) with p = 2 gives for any k = 0 , 1 , . . .

Proposition A.11 gives for any k = 0 , 1 , . . . the bound

which implies

We set

The continuity of probability measures and Lemma A.12 then imply

Hence, if then,

where we used that we can assume d ≥ 2 without loss of generality. Additionally, Eq. (43) and the elementary inequality 1 -x ≤ exp( -x ) , which is valid for any real number x, give

When d = 1 , the right hand side of this inequality is 0 . For d ≥ 2 , we can also use the bound d -1 ≥ d/ 2 . Together with

this concludes the proof of the first statement. For the proof of the second statement, we apply Markov's inequality to obtain

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this manuscript, the authors return to classic pair-based STDP models, in which the change of a synapse depends on the relative timing of the spikes of the pre- and postsynaptic neurons. The authors prove mathematically that this type of STDP model identifies the pre-neuron with the highest triggering probability. Moreover, they show that it is connected to noisy mirror descent.\n\nSTRENGTHS AND WEAKNESSES:\nIn the literature (reviewed in the cited paper of Morrison et al.), there are many versions of such a pair-based STDP rule. If I understand correctly, the authors focus on the one where, for the sequence post-before-pre and pre-before-post, only the closest postsynaptic spike matters, whereas all presynaptic spikes between two postsynaptic spikes are taken into account. This assumption is well accepted, and I don't criticize it.\n\n\nExisting analytical models of STDP (e.g. Kempter et al. 1999 as cited) have mostly assumed stochastic spike arrival with Poisson rate $\\lambda_j$  and stochastic output spiking with instantaneous rate (conditional stochastic intensity)  $\\lambda_{post} = \\sum_j w_j \\int_0^\\infty \\epsilon(s) \\lambda_j(t-s) ds$ with some integration kernel $\\epsilon$. They found that pair-based STDP performs PCA. For uncorrelated Poisson input, there is a positive feedback loop proportional to the overlap\n$\\int_0^\\infty  W(s) \\epsilon(s) w_j \\lambda_j ds$ where $W(s)$ is the causal (=pre-before-post) section of the STDP window.\n\nThe authors modify this standard paradigm a little bit in that they assume that the postsynaptic neuron has no intrinsic noise. Instead, output spikes are defined by a deterministic threshold crossing process (leaky integrate-and-fire, LIF). Under the assumption (i) that the kernel $\\epsilon$ is a jump followed by exponential decay, and (ii) that the stochastic firing intensity of presynaptic neurons is constant, the authors correctly observe that the threshold crossing occurs at the moment of (or arbitrarily shortly after) presynaptic spike arrival and that the relative probability that a presynaptic neuron j is the one that triggers the postsynaptic spike is (if all weights are small)\n\n$$p(j) = \\frac{\\lambda_j w_j}{\\sum_l \\lambda_l w_l}.  (Eq. 12)$$\n\nI like this observation, and it looks novel to me!\nFrom this observation, together with the assumption that a sequence pre-before-post leads to weight increase, it follows immediately that the input weight vector becomes in the limit parallel to $p(j)$. Indeed, the resulting self-feedback term is very close to that of Kempter et al. (1999). The difference is that in the noisy neuron model of Kempter, the full time course of $\\epsilon$ plays a role, whereas in the deterministic LIF model, only the initial moment of the rapid rise of $\\epsilon$ plays a role. Indeed, it is known in computational neuroscience that the threshold process of an LIF generates correlations that are sensitive to the derivative of $\\epsilon$.\n\nStarting from Eq. 12 (which is unfortunately only given in Section 2.3), the authors have arrived at a formal mathematical model that distinguishes the input spike that triggers the postsynaptic spike from all other input spikes. This model enables mathematical analysis. As expected from earlier work, weights explode, and the weight vector becomes aligned with the strongest input.\n\nIn the formal mathematical model (from line 225), the algorithm \"selects a presynaptic spike time $\\tau^* > t_k$ and then sets $t_{k+1} = \\tau^∗$. This is done in two stages by first choosing a presynaptic neuron $j^∗(k + 1) \\in [d]$ and, in a second step, picking a spike time $τ^∗$ among the spike times of the $j^∗(k + 1)^{th}$ presynaptic neuron.\" I would have preferred to see first the biological motivation of Section 2.3. before the abstract mathematical model, but this is a matter of taste and only a minor point.\n\n----\n\n**Major**:\n\n1. The authors are not aware of important closely related literature  which limits the novelty of their results. Between 1996 and 2007, (pair-based) STDP attracted a lot of research interest.\n\n1.1 Kempter et al. (1999, cited) already analyzed the evolution of synaptic weights in a mathematical model where the (stochastic) generation of output spikes depend linearly on the input spikes. Even though the noise model is different from that of the current manuscript, the results are essentially the same. Importantly, Kempter et al. (1999) cover a much broader range of input paradigms that go well beyond the independent homogeneous Poisson processes considered in the current manuscript.\n\n1.2. As part of the efforts 20 years ago, several loss functions have been proposed that yield STDP via gradient descent. Noteworthy examples are Chechik 2003, Bohte and Moser 2005, Bell and Para 2005, Toyoizumi et al. 2005; Toyoizumi et al. 2005a, Pfister et al. 2006, Toyoizumi et al. 2007.\n\n1.3. The general theoretical framework of gradient descent in computatonal neuroscience has been reviewed by Surace et al. 2020 who highlight the importance of a metric in order to fully define gradient descent. The metric of mirror descent is one example. A rescaling of parameters from a to b=exp(a) to guarantee positive values of b is a standard trick in the field.\n\n2. It is unfortunate that the authors consider only a very specialized stimulation paradigm and a very specific case of STDP.\n\n2.1. I believe it would be possible to generalize the approach (i)  from homogeneous to inhomogeneous Poisson input using the time rescaling theorem and (ii) from independent to correlated Poisson inputs. For the latter, several possibilities exist to generate correlations in the input. Note that Kempter et al. (1999) already analyzed for their neuron model a rather general case of inputs. They indeed found that STDP generated PCA as conjectured in the manuscript.\n\n2.2 In standard pair based STDP there are four parameters, i.e., two time constants and two amplitude parameters. There is a well-known special case of STDP which is called balanced STDP. In this special case the total area under the STDP window is zero. If, in addition you require that the time constants for pre-before-post and post-before-pre are identical, then we get an exactly antisymmetric STDP window.  If I am not mistaken, the authors make both assumptions (integral zero and same time constant) to get their Eq. (11). How would the results generalize to other cases?\n\n2.3. Unfortunately, pair-based STDP is NOT a good representation of biological data as shown by Sjostrom et al. (2001). First, more than two spikes interact ('triplet effect') and second, the voltage at the location of the synapse also plays a role (for both, see e.g.  the cited paper of Clopath et al). These observations  might have contributed to the reduced interest in computational neuroscience for standard pair-based STDP models even though they are still widely used.\n\n\nAdditional references:\n\n\nSjostrom, P., G. Turrigiano, and S. Nelson (2001). Rate, timing, and cooperativity jointly determine cortical synaptic plasticity. Neuron 32, 1149–1164.\n\nChechik, G. (2003). Spike-timing-dependent plasticity and relevant mututal information\nmaximization. Neural Computation 15, 1481–1510\n\nBohte, S. M. and M. C. Mozer (2005). Reducing spike train variability: A computational\ntheory of spike-timing dependent plasticity. In L. K. Saul, Y. Weiss, and L. Bottou\n(Eds.), Advances in Neural Information Processing Systems 17, pp. 201–208. Cam-\nbridge, MA: MIT Press\n\nBell, A. J. and L. C. Parra (2005). Maximising sensitivity in a spiking network. In L. K.\nSaul, Y. Weiss, and L. Bottou (Eds.), Advances in Neural Information Processing\nSystems 17, pp. 121–128. Cambridge, MA: MIT Press.\n\n\nT. Toyoizumi, J.-P. Pfister, K. Aihara, and W. Gerstner (2007)\nOptimality Model of Unsupervised Spike-Timing Dependent Plasticity: Synaptic Memory and Weight Distribution\nNeural Computation, 19: 639-671\n\n\nT. Toyoizumi, J.-P. Pfister, K. Aihara, and W. Gerstner (2005)\nGeneralized Bienenstock-Cooper-Munro rule for spiking neurons that maximizes information transmission\nProc. Natl. Acad. Sci. USA, 102:5239-5244\n\nToyoizumi, T., J.-P. Pfister, K. Aihara, and W. Gerstner (2005a). Spike-timing dependent\nplasticity and mutual information maximization for a spiking neuron model. In L. K. Saul, Y. Weiss, and L. Bottou (Eds.), Advances in Neural Information Processing Systems 17, pp. 1409–1416. Cambridge, MA: MIT Press.\n\nJ.-P. Pfister, T. Toyoizumi, D. Barber, and W. Gerstner (2006)\nOptimal Spike-Timing Dependent Plasticity for Precise Action Potential Firing in supervised learning\nNeural Computation 18:1309-1339\n\nT. Toyoizumi, J.-P. Pfister, K. Aihara, and W. Gerstner (2007)\nOptimality Model of Unsupervised Spike-Timing Dependent Plasticity: Synaptic Memory and Weight Distribution\nNeural Computation, 19: 639-67\n\nS.C. Surace, J.-P. Pfister, W. Gerstner, and J. Brea (2020)\nOn the choice of metric in gradient-based theories of brain function\nPLoS ComputBiol 16(4):e1007640,\ndoi: 10.1371/journal.pcbi.1007640\n\nQUESTIONS:\nPlease see \"Strengths And Weaknesses\" for my concern and questions.\n\nLIMITATIONS:\nNo. The authors have not adequately addressed the limitations of their work. While the theoretical analysis is presented clearly, the manuscript overlooks several critical limitations that should be acknowledged and discussed:\n\n- Connection to Prior Work: The authors are not aware of important, closely related literature, which limits the novelty of their results. (Please see \"Strengths And Weaknesses\" for more detail.)\n\n- Narrow Scope of Input Models: The work assumes independent, homogeneous Poisson inputs—a highly restrictive and biologically unrealistic setting. This limits the applicability of the results. The potential for generalization to inhomogeneous or correlated inputs should be addressed.\n\n- Simplifying Assumptions in STDP Rule: The analysis is restricted to a specific, highly idealized form of pair-based STDP—likely the balanced, antisymmetric case. The authors should clarify whether their results hold for more general STDP methods.\n\n- Biological Plausibility: The model assumes pair-based STDP, which has been shown to be an incomplete description of biological plasticity. Experimental studies (e.g., Sjöström et al., 2001; Clopath et al.) have demonstrated the importance of triplet interactions and voltage dependence. These factors should at least be acknowledged as limitations of the modeling framework.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this work, the authors establish a mathematical connection between spike-timing-dependent plasticity (STDP) and noisy gradient descent on a probability simplex. The nature of this connection allows for mathematical analysis which goes beyond previous techniques which utilized ensemble averages. The main finding of their analysis is that their STDP learning rule converges exponentially fast to identify the presynaptic neuron with the highest firing rate. Additionally, they establish a connection to noisy mirror descent, specifically entropic gradient descent.\n\nSTRENGTHS AND WEAKNESSES:\nThis work provides a novel analysis of STDP, one that is certainly significant to the fields of computational neuroscience and AI. The mathematical formulation linking local spike-based updates to global optimization is valuable. The  mirror descent perspective is of particular interest, connecting STDP to an intriguing framework from Cornford et al. 2024. The assumptions and proofs are well described, though it may benefit the authors to target a slightly wider audience. While the analysis is novel and interesting, it does require some assumptions which limit its applicability to STDP in biological conditions (the authors acknowledge these limitations).\n\nQUESTIONS:\nRegarding the selection probabilities (section beginning line 238), I would assume that, under most biological conditions, a handful of presynaptic spikes (on the order of tens) would be sufficient to trigger a postsynaptic spike. Is there any perturbation or corrective term which can be added to move even slightly away from the two limit cases (“weights are very small” or “weights are large enough to for any presynaptic activity to trigger a postsynaptic spike”)? \n\nI lost the plot a bit around theorem 2.2. I was able to answer the questions I had by going to the appendix, but it broke up the flow of the paper a bit for me. Perhaps the authors can consider adding more “connective tissue” in the main text to guide the reader, some bits are currently limited to an expert audience.  \n\nI would be curious to see a more direct comparison to the previous methods of STDP analysis which used ensemble averages.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies spike-timing dependent plasticity (STDP) in a feedforward network of d input neurons projecting on a single postsynaptic neuron. Rather than analyzing  the classical STDP model (Gerstner 1996), which the authors argue is complex to analyze, the authors consider a simplified version of it, which is described in section 2.3. The authors make a thorough analysis of this simplified model, showing convergence by casting its dynamics as noisy gradient descent of a specific loss function. Notably, this simplified model behaves as a winner-take-all selection mechanism, with the postsynaptic neuron aligning with the highest firing input neuron. This is a departure from the rate case (e.g. Oja’s rule) in which the post-synaptic neuron aligns with the first principal component of the input layer.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- The math and presentation of the derivations are very rigorous and well-explained.\n- Convergence of this model is rigorously established by casting it as noisy gradient descent.\n\nBroadly, the paper was a valiant effort at using sophisticated mathematical tools to analyse a system of neuroscientific interest. It performs a thorough theoretical analysis of a simplified model that is appreciated.\n\nWeaknesses:\n\nHowever, the value of the contribution is hindered slightly by the clarity of the presentation for neuroscientist, but majorly by the large leap required between STDP and the stated model.\n\n- Relevance of the simplified model to neuroscience: The jump from the classical STDP type model to the simplified model is worrying. A crucial part of the integrate-and-fire model is the fact that it integrates inputs in order to choose when to spike. On first reading, this appears to be completely missing from the simplified model. Further, the classic components of STDP, such as spike-time dependent kernels are also missing. These details all seem to be wrapped up in the procedure for selecting the presynaptic spike at which the postsynaptic neuron fires. However, this seemingly crucial selection procedure is somewhat swept under the carpet (line 236). The best version of this claim would be that you can choose this selection procedure in order to model a classical integrate-and-fire neuron with STDP. Yet no evidence is presented that this is possible. Providing such evidence would significantly strengthen the article. More broadly, more discussion regarding these differences between the toy and classical model is strongly recommended.\n\n- High level presentation: The paper seems formatted backwards in some ways. For example, the easier to understand analysis, on gradient flow, arrives after the complex discrete time analysis. Further, the movement from the classical integrate-and-fire model to the more toy model was presented after a long discussion of the toy model. For a neuroscientist the natural place to start would be the classical model, then explain the simplifications, before ending with the resulting analysis of the model.\n\nMore minor points:\n\n- Some figures are not informative: e.g. Fig 1: shows a feedforward network of spiking neurons, which is very common in the field, but gives no insights on the plasticity rule modeling. A cartoon would greatly help to ease the reader into the authors’ way of considering STDP.\n\n- The links to mirror descent could be interesting, but it is unclear how it is used to create actionable insights. It feels like it is another mathematical framework presented regarding dynamics that have the same equilibria, and the neuroscientifically interesting conclusion is the structure of these equilibria. What then do we gain via an alternative characterization of the dynamics? It would be good to signpost the role it is hoped that this additional analysis plays.\n\n- Typo in the appendix: the loss on line 769 is missing a square.\n\nQUESTIONS:\n1/ Could the authors explain how their simplified model incorporates the integration behavior of standard STDP? E.g. how are the postsynaptic spike times dependent on the threshold S?\n\n2/ A classic result for rate models, that the authors point to, is that rate models under with Hebbian plasticity learn to extract the principal components of the input data. Often there is a somewhat natural mapping between spike and rate models, specifically STDP and Hebbian learning, that would bridge the existing PC results to the authors’ results regarding the alignment of the output neuron to the highest firing input neuron direction. This is interesting. Do the authors have any suggestions about why that does not appear to be the case here?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors provide a new theoretical analysis of the convergence behavior of Hebbian STDP rules. First, they show that a Hebbian STDP rule can be reformulated as gradient descent on a loss function defined over the presynaptic neurons spike-triggering probability simplex. Then, they prove that such spike-triggering probability converges to the basis vector of the probability simplex pointing to the presynaptic neuron with highest initial spike-triggering probability. Lastly, they discuss connections to noisy mirror descent.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: The mathematical derivations are theoretically sound. The rigorous convergence analysis of p(k) in STDP is a potentially important theoretical result. Viewing STDP as gradient descent on the presynaptic neurons' spike-triggering probability simplex is, to the best of my knowledge, a novel and compelling idea that may inspire further theoretical developments.\n\nLimitations:\nDespite its mathematical rigor, the manuscript lacks clarity in key areas, and in its current form, the novelty does not appear sufficient to meet the standards expected at NeurIPS.\nFirst, it is unclear whether the authors introduce a new Hebbian STDP rule or analyze a family of existing ones. Lines 12–13 state they “provide an analysis of a STDP rule,” while line 83 says they “introduce a STDP rule,” suggesting novelty without clarification. Yet, Section 2.3 presents a biologically plausible STDP rule that has already been studied in the literature (e.g., Ref. 11), and that can be reformulated to match Eq. (1). This suggests that Eq. (1) generalizes a known class of STDP rules, with Eq. (11) as a specific instantiation. However, the manuscript does not clearly state which rules fall under this general form.\nThe paper's structure further obscures this point: Eq. (1) is introduced early, without sufficient context, and its connection to established STDP formulations only emerges later, in Section 2.3. This non-linear narrative makes it difficult to follow the paper’s logic.\nThe authors should more precisely specify the generality of Eq. (1) in relation to known STDP variants that go beyond the brief description in lines 27–28. Experimental studies support the existence of multiple types of STDP rules, varying across brain regions and cell types (Fino et al. J Physiol 2008, doi: 10.1113/jphysiol.2007.144501; Perez et al. Cell Rep 2022, doi: 10.1016/j.celrep.2022.110521). While this diversity is not yet fully leveraged in SNNs, the authors should acknowledge it and specify the generality of Eq. (1). \nIt is also unclear what results could be derived using the previous methods mentioned in lines 35–37, and which results uniquely stem from the gradient-descent interpretation introduced here.\nWhile the derivations appear theoretically sound, their novelty and significance remain uncertain. As the authors also point out, convergence to a winner-take-all regime has been previously observed for Hebbian STDP models. The authors do not sufficiently discuss the added value of their convergence result, which appears to be the main theoretical contribution of the paper. The connection to noisy mirror descent is also only briefly mentioned and not clearly motivated or explored. \nOverall, proving the convergence of a specific family of STDP rules to an already known regime – within the simplified setting of a single-readout neuron - constitutes, in my view, a limited theoretical contribution that does not meet the standard expected at NeurIPS. Rigorously extending their theoretical result to the scenario of an arbitrary number of readout neurons would significantly strengthen the contribution.\nMinor comments:\n•\tEq. (1): The quantity Z(k) should be defined immediately after the equation. Also, explicitly stating that k indexes postsynaptic spikes would aid clarity.\n•\tThe writing style should be improved throughout. For instance, the sentence “We continue by relating the learning rule (3)” (line 101) appears before Eq. (3) is introduced and is difficult to follow.\n•\tTo avoid confusion, I recommend consistently prefixing equation references with “Eq.” to distinguish them from citation numbers.\n\nQUESTIONS:\nCurrently, the argument in Section 5 of the paper is not rigorous enough. Can you prove convergence of p(k) for an arbitrary number of readout neurons?\n\nCan you provide an intuitive interpretation for the loss function in Eq. 5?\n\nWhich is the practical significance of Theorem 2.2 for training SNNs with STDP rules?\n\nWhich known STDP rules fall under the formulation of Eq. (1), and which do not?\n\nLIMITATIONS:\nCurrently, it is not clear which STDP rules belong to the Eq. (1) family. The authors should specify that Theorem 2.2 is limited to STDP rules belonging to the family in Eq. (1).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}