Meta-learning families of plasticity rules in recurrent spiking networks using simulation-based inference
Authors: Basile Confavreux
Poornima Ramesh
Pedro J. Goncalves
Jakob H. Macke
Tim P. Vogels
Venue: NeurIPS 2023 poster (2023)
Keywords: synaptic plasticity
spiking network
meta-learning
computational neuroscience
TL;DR: We introduce a flexible approach for principled exploration of complex plasticity rules in large recurrent spiking networks, enabling more robust predictions and thus deeper insights into the plasticity mechanisms underlying brain function.

Abstract: There is substantial experimental evidence that learning and memory-related behaviours rely on local synaptic changes, but the search for distinct plasticity rules has been driven by human intuition, with limited success for multiple, co-active plasticity rules in biological networks. More recently, automated meta-learning approaches have been used in simplified settings, such as rate networks and small feed-forward spiking networks. Here, we develop a simulation-based inference (SBI) method for sequentially filtering plasticity rules through an increasingly fine mesh of constraints that can be modified on-the-fly. This method, _filter SBI_, allows us to infer entire families of complex and co-active plasticity rules in spiking networks. We first consider flexibly parameterized doublet (Hebbian) rules, and find that the set of inferred rules contains solutions that extend and refine -and also reject- predictions from mean-field theory. Next, we expand the search space of plasticity rules by modelling them as multi-layer perceptrons that combine several plasticity-relevant factors, such as weight, voltage, triplets and co-dependency. Out of the millions of possible rules, we identify thousands of unique rule combinations that satisfy biological constraints like plausible activity and weight dynamics. The resulting rules can be used as a starting point for further investigations into specific network computations, and already suggest refinements and predictions for classical experimental approaches on plasticity. This flexible approach for principled exploration of complex plasticity rules in large recurrent spiking networks presents the most advanced search tool to date for enabling robust predictions and deep insights into the plasticity mechanisms underlying brain function.

Meta-learning families of plasticity rules in recurrent spiking networks using simulation-based inference
Basile Confavreux*
Poornima Ramesh University of Tübingen, Germany pooriimarraaeee1111801..a. com
Institute of Science and Technology Austria basile.confa cestlee.cofaarruuggaal..cc
Pedro J. Gonçalves
University of Tübingen, Germany
VIB-Neuroelcctronics Research Flanders (NERF), Belgium imec, Belgium
Jakob H. Macke t University of Tübingen, Germany Max Planck Institute for Intelligent Systems, Germany
Tim P. Vogels Institute of Science and Technology Austria
Abstract
There is substantial experimental evidence that learning- and memory-related behaviours rely on local synaptic changes, but the search for distinct plasticity rules has been driven by human intuition, with limited success for multiple, co-active plasticity rules in biological networks. More recently, automated meta-learnig approaches have been used in simplified settings, such as rate networks and small reed-forward spiking networks. Here, we develop (SBI) method for sequentially filtering plasticity rules through an increasingly fine mesh of constraints that be modified on-the-fly. This method, filter SBI, allows us to infer entire families of complex and co-active plasticity rules in spiking networks. We first consider flexibly parameterized doublet (Hebbian) rules, and find that the set of inferred rules contains solutions that extend and refine-and also reject-predictions from mean-field theory. Next, expand the search space of plasticity rules by modelling them as multi-layer perceptrons that combine several plasticity-relevant factors, such weight, voltage, triplets and co-dependency. Out of the millions of possible rules, we identify thousands of unique rule combinations that satisfy biological constraints like plausible activity and weight dynamics. They can be used as a starting point for further investigations into specific network computations, and already suggest refinements and predictions for classical experimental approaches on plasticity. This flexible approach for principled exploration complex plasticity rules large recurrent spiking networks presents the most advanced search tool to date for enabling robust predictions and deep insights into the plasticity mechanisms underlying brain function.
iouuulation-bsee inference
1 Introduction and related work
Synaptic plasticity has received continuous attention over the past decades 15. how functions such as learning and memory emerge from local synaptic changes is still poorly understood. Given the experimental inaccessibility of synapses vivo, many studies have relied
Nevertheless,
* 4: equal contributions
37th Conference on Neural Information Processing Systems (NcurIPS 2023)

---

on theoretical methods to shed light on the plasticity rules at play 12] However, this approach does not scale B in particular, previous attempts tuning complex, co-active plasticity rules have been limited by human intuition, and had to rely on theoretical frameworks based on strongly simplifying assumptions
Recently several studies have aimed to address these issues by automating the discovery of plasticity rules using numerical meta-learning approaches (0) Given the non-differentiabiiity of biological systems, along with the steep compute requirements of simulating plastic networks at scale, these studies have been restricted rate networks 13
16 20 or small feed-forward spiking networks
15 [7]. Moreover, these approaches require the priori definition and optimization of loss function that enforces desired network computation, as well as various regularizers to ensure the implementation of the task is biologically plausible. Lastly, the above studies typically propose single plasticity mechanisms compatible with the data or network function considered.
Here, we aim at a more comprehensive approach to discovering biologically plausible plasticity rules. We study network dynamies large recurrent spiking networks with plasticity rules that are flexibly parameterized (using polynomials or neural networks, Fig. [A]. We introduce filter simulation- -ased inference (ISBI), a new meta-learning approach that can thoroughly explore the space of potential rules. This method allows us to select sets of rules using metrics that effectively constrain network dynamies to fulfill the desired conditions. Our approach successfully infers entire families of rules that robustly establish plausible dynamics in large spiking networks. We demonstrate the scalability of ISBI by applying it to neurrllnetwork-baaee to influence plasticity, such as, e.g., synaptic weight or membrane potentials. Despite the nonlinear interactions induced by the neural network parameterization of plasticity and the increase in the number of factors, fSBI successfully recovers plausible rules. Interestingly, when considered in isolation, the inferred candidate rules often result in contradictory experimental predictions, revealing that the classical pre-post protocols may only have marginal explanatory power.
search spaces that include additional factors known
Overall, our study introduces a principled approach for meta-learning plasticity rules in complex networks. The inferred distributions of plausible rules can be used as starting points for future studies of specific network computations, and raise questions about classical theoretical and experimental protocols for understanding plasticity.
A Plestic recurrent spiking network
B
Prior <x(0)
Filtered posterior p(0|teuc Tea) (8)
Filtered posterior 2 p(Oläw) = xy(0)
Parameterized plasticity rules
Ae flocal vars, 8)
O refine
O reline
Polynomial
MLP
Iteratively narrow-down the search-space
Figure 1: Flexible inference of plasticity rules in recurrent spiking networks with fSBI.
A: Plastic recurrent spiking networks are simulated with parameterized plasticity rules (parameters f), either as polynomials (Figs. 2 3 or as multi-layer perceptrons (Fig. 4. B: Filter-SBI (fSBI). Starting from a prior over plasticity parameters To (0), we infer a posterior given a first network metric, for instance the population firing rate, pı (0) rexe O rimb). ). In contrast to other classical Bayesian inference approaches, n = P1 (0) rexe O rinh) used as a prior for the inference in the subsequent round, given different network metric. We then repeat this process for various network metrics.
2 Methods
2.1
Spiking network model
We consider a recurrent spiking network 4096 excitatory and 1024 inhibitory leaky integrate-and- fire neurons with conductance-baset synapses, relative refractoriness, adaptive threshold and NMDA currents. The model parameter values follow Zenke et al. [8] (Fig. [A; see Supp. 6u for a complete description of parameters). The membrane potential dynamics of neuron j (excitatory or inhibitory) are given by
dV, 5m dt
-V, Vrea)) ())() (V, Ex) g)()) (V, - -E),
(1)
2

---

where E stands for excitation and for inhibition. A postsynaptic spike is emitted whenever the membrane potential (t) crosses threshold (t), with an instantaneous reset to Vesst- This threshold y (t) is incremented by With spike every time neuron j spikes and otherwise decays following
d J
1-th basc
V
Tth The excitatory and inhibitory conductances, gr and gt evolve such that
dt
dgf
g3
(1() = agggn// (t) + (1 a)gJ (1) and
(1)
+
(is(t)d;(t)
di
TGABA
iEInb
(3)
d
AMAMA
9
AMPA
dg
NMDAA
AMPA
9)
NMDA
with
and wiy (t)(())))
dt
TAMPA
dt
TNMDA
KERx
with Way (t) the connection strength between neurons and (unitless), (z(t) Σ 8(t (x) the spike train of pre-synaptee synaptic neuron k, where & denotes the spike times of neuron kt and the Dirac delta. All neurons received input from 5k Poisson neurons, with 5% random connectivity and constant rate Text randomly sampled from the uniform distribution (1, 15) each simulation. The recurrent connectivity was instantiated with random sparse connectivity (10%).
2.2 Synaptic plasticity parameterizations
In order to explore a wide range of possible plasticity rules, we defined parameterized functional forms for these rules (i.e., search spaces) enabling us to flexibly generate various rules, by changing the parameters of the function. We considered two such parameterizations:
For polynomial plasticity rules, each recurrent synapse underwent spike-timing dependent plasticity (STDP) (21) The weight from neuron i (of type X and Y; (X, Y) E (E, 1)) evolved such that
dwi 1) (() + (x) (5x)) + (x)),
(4)
di
with 7) fixed learning rate. The variables v. (1) and z (4) trace the pre- and post-synaptic spike trains
dæ;
5,(1) and dæ dt
T XY
+ (1),
di
where TXY and TXY are the time constants of the traces associated with the pre- and postsynaptic neurons, respectively. Thus, the plasticity updates each connection type depended on parameters: ppss J. for total of 24 parameters across all connection types OXY [axY, Bxy, 7XY, KXY; :x:: TX (Fig. A).
prre post
For the MLP plasticity rules, only the recurrent EE and IE synapses underwent spike-triggered updates (Fig. [4)). Change the weight from neuron to j (of type X and E, X E (E, D)) was computed by rumning forward multi-layer perceptrons (MLPs) that took inputs several synaptic variables relevant to plasticity! m
dwis dt
(t) (1. (1,3, XF
, Wijs (V)), CTCC c(igh)) +
(1)
(6)
(t)
(1(1), II Wiy, (V)), O5X0 Clan)) )
(2)
XE
with Cyte= = (o (Fg 1o.--V)) V2)) and Cjah- = (o) (Fr - V2)) co-dependent terms representing the activity of neighboring synapses [00 (here all synapses with the same postsynaptic neuron), and (V)(t) the low-pass filtered membrane potential [T]. In this search space, time constants of synaptic traces were (1) = TI.
We chose wwo-hideen-layer MLP, composed 50 sigmoidal units the first hidden layer, and the second. We trained only
fixed at TE
(1) = 10ms and TE
(2) = TTE 100ms (22 7 g 10).
(2)
the final layer, keeping all other layers fixed. While this is standard practice with pre- pre-trained ML.
Note that all input variables the MLP are functions of t, which elide lighten notation.

---

models, we randomly initialised the parameters of the frozen layers, rather than pre- pre-training them. We verified that when using random initializations for the first 2 hidden layers (~ u -1 Thee" where Vimp is the number of input features at a given layer), and only training the weights, bias, and output learning rate of the final layer, previously observed plasticity rules could be approximated (Supp. (6.7).
given plasticity rule (EE or IE) thus consisted of one MLP (with 5 parameters: weights and bias) to update the synaptic weight when a pre-spike occurred, another MLP updating the synaptic weight for post-spikes (Fig. 43), and a common learning rate 7XE for both MLPs, resulting in 11 parameters per rule, and a total of 22 parameters between EE and IE.
2.3 Mean-Field predictions
In the polynomial case, we performed self-consistent analysis of the weight and activity dynamics in the network at steady state, following previous work [23] [0] (more details for all connection types in Supp. 6.3). For example, EE synapses, could calculate expected firing rate rex from the parameters of the rule such that:
SEE
Texc
λEE
with λEE = KEETEE
post pre
YEETER
2.4 Assessing network plausibility with metrics
To assess whether a particular plasticity rule was suitable candidate, we simulated a spiking neuronal network (SNN) with that rule and quantified the resulting dynamics with multiple metrics on the activity and weight traces. The metric choices and values were informed by-and in line with range of modeling and experimental cortical studies [7 g 24 For every simulation, we stored the raw spike trains and weight traces for 10s after 1 10s of simulated time (60s for the MLP rules). We grouped 15 individual metrics into four main criteria:
Stable activity dynamies: Rules were flagged as suitable when the excitatory and inhibitory popula- tion firing rates were bounded between I and 50 Hz, i.c., Texe E [1, 50) Hz and rinh E [1, 50 Hz, in line with large swathe of literature reporting cortical firing rates of few Hertz on average 24 29 30
Stable weight dynamics: Fraction of synaptic weights that reached extreme values (0 or Wmix),
fuww: mean absolute change in synaptic weights between simulation start and end across all synapse types, Wereen mean weight at the last time-step across all synapses of a specific type, (wxy). These metrics excluded rules whose weights changed too rapidly, did not converge, or reached unrealistic values. Rules were flagged as suitable if they satisfied f < 0.1, Wereep < 0.05, (WEX) < 0.5 and (wxx) < 5 simultaneously.
Near-irregular dynamics: Mean coefficient of variation of the inter-spike intervals across all excitatory neurons, (cv(ISI,)); auto-covariance for each spike-train, and its integral, averaged over time and the neural population, (p),,; Fano factor for each spike train averaged over the population, (Fano);; standard deviation of the firing rate of individual neurons averaged across the population, (o); These metrics quantified the regularity of a spike train, but, either due to metric assumptions or hyperparameter choices, each metric exhibited different failure mode, which could be filtered out by combining the metrics. Rules were flagged as suitable if they satisfied (cv(ISI:)) > 0.7, (phi, < 0.1, (Fano); [0.5, 2.5) and (a); < 5 simultaneously.
Near-asynchronous dynamics: Standard deviation of the excitatory population firing rate, r the Fano factor averaged across the population and then across time windows, (Fano) we also computed the population average of binned spike-trains, power spectrum across all frequencies f (except 0), < S >i.. Rules were flagged as suitable if they satisfied (Fano); E 2.5), < S >i,5< and or.. < 0.05 simultaneously. These bounds enforced the broadly asynchronous iiregular activity observed in cortex (2] 25 27 29 30). Ranges for metrics assessing synchrony/regularity were thus devised with independent Poisson spike trains in mind, with leeway for spatiotemporal correlations, as seen in cortical recordings (32
trrrier-trasfformed it and averaged the resulting
Rules were considered plausible if they fulfilled all four broad criteria above (more details and analysis of individual metrics in Supp. 6.2

---

2.5 Filter ioauuaation-basee inference
Having defined plasticity rules search spaces and network metrics, we turned to the problem of exploring these spaces, systematically filtering out unsuitable rules.
approach for performing Bayesian inference given stochastic models (with tunable parameters 0, and generating samples x) in cases
Simulation- inference (SBI) is broadly defined
where the likelihood of the model p(a: (0) is analytically inaccessible but simulation is possible [35
37.. SBI infers the posterior distribution p(0]zz), using samples from a prior distribution 0 ~ π (0)) and corresponding simulations x ~ p()) i.e., it applies Bayes' rule such that p(0|x) x p(a ()) T(0).
where p(0 2z) represents the distribution over model parameters most likely to have generated x.
This is appealing framework for the problem of identifying plausible plasticity rules: we treated the parameters of the plasticity rule as parameters 8 of a stochastic plastic spiking network that generated simulations x consisting of the metrics defined in Section 2.4 We then used SBI- e.g..
Neural Posterior Estimation (NPE) 35 38 39 to identify the posterior distribution over plasticity rule parameters that were most likely to have resulted in plausible network dynamics. However, could not know beforehand which parameters were most likely to establish plausible dynamics, we had to define broad prior over the parameters of the plasticity rule (f)) In turn, the broadness of the prior meant that potentially millions of SNN simulations would be needed to find a few plausible rules, rendering this search method computationally unfeasible [40
In order to address this issue, we developed filter SBI (fSBI). In ISBI, we applied the metrics quanti- fying plausibility sequentially, rather than all once (Fig. [ Towards this goal, we first sampled plasticity rule parameters 9 from a uniform prior and simulated an SNN with the corresponding rules. Next, computed the various metrics defined in Section 2.4 We then used NPE to fit posterior density over the plasticity rule parameters. Finally, we sampled all parameters that satisfied a given criterion from the posterior, e.g., stable network activity p(f]rexc E [1, 50) Hz O rinh [1, 50 Hz), in the first instance, effectively "filtering out" other rules from the prior. These posterior samples were then used as the prior samples for the next round of inference, in which an additional criterion was applied to obtain a new, narrower posterior. Mathematically speaking, a filtering round k using metrics m& fa (x) and corresponding conditions gx(mm)) proceeds as follows:
(f) Pa-1 (0) m 8.1. ge- (mx-1))
(8)
(1(0 < p(mp|0) (0)
(9)
1 (0) pa (0)me s.l. Igg. (mz)),
(10)
where T is an indicator function over the condition g. This process is repeated until only plausible candidate rules remain, for total of around 200k simulated rules per search space.
The distribution obtained at each filtering round is not necessarily true Bayesian posterior, since (a) we change the metric on which we condition the posterior at each round, (b) we sample the posterior for a range of metric values rather than a single value, and (c) we also do not correct for the fact that each round, we are sampling from the posterior of the previous round rather than from the prior (more detailed analysis of this "pseudo" moadddo-posteer "posterior in Supp. 6.5).
42
3 Results
In our attempt to discover plasticity rules compatible with given function, we introduced ISBI, a meta learning method that sequentially filters out unfit candidates rules from vast sets of possible plasticity rules in large recurrent spiking networks (SNNs, Methods. 2.1 TO). We first applied fSBI in the polynomial search space described in Section 2.2 and compared the resulting distributions of rules to those predicted by mean-field analysis. We then applied ISBI to plasticity rules which themselves were parameterized by neural networks. In both cases, iSBI inferred expected and novel rules that robustly establish plausible dynamics in SNNs, while also eliminating some theoretically viable solutions.
3.1 ISBI infers plausible rules from polynomial search space
We initially considered 4 co-active plasticity rules, E-to-E (EE), E-to-I (EI), I-to-E (IE) and I-to- I (II) in a recurrent spiking network of 5120 conductance-baat based leaky integrate-and-fire neurons (Fig. A). The network architecture and parameters were chosen following Zenke et al. [8]. Each
5

---

A Awwr(()) Poly(xpre--xpot
B
Stable weights
fraction of rules sampled from
1
stable activity
(W (twes()) (i
A.I.
10"
stable weights
Stable activity
10
EXC
Inh
10-3-
A.I. dynamics
(0)
-10
plausible
Text
No
11
112
11o
1
12
n1
D
E
Example plausible rule from
explained
variance
0.10
Exc
Text
5lzx
Inh Exc
Text 15Hz
AWEE
0.05
11
122
Inh
-0.2
0.2
0.00
fTa
AWai
@ dimensions
24
(cv(iSI))
1
1000 (H2)
10
10
At
10 neurons
10 neurons
0.1
'Wee "Wei
0.1
-Woo We
Aww Awe
-0.2
Ac (s)
2
"Wi *Wa
2
-Wi "W.
-022
0.2
0.5
0.5
100 ms
100 ms
55
t (s)
(s)
60
AC (s)
Figure 2: Posterior of plausible plasticity rules with fSBI. Polynomial search-space.
A: Recurrent spiking network receiving Poisson inputs at a random rate Text, with each recurrent connection parameterized with an STDP plasticity rule. Dotted lines denote plastic connections. B: Starting from uniform draws (TT) from the polynomial search space, fSBI progressively filtered out the rules that do not establish plausible dynamics in an SNN (T., 22. T3), A.I. stands for asynchronous irregular. C: Fraction of rules sampled from each intermediate iSBI posterior that obey conditions on the corresponding network metrics. D: Principal component analysis (PCA) on samples from each intermediate ISBI posterior (To + Tg). Variance explained along the corresponding PCA directions across rounds. E: For each plastic network simulated in fSBI (pooled across all rounds), the corresponding CV and covariance metric: network fulfills the auto-covariance condition (p < 0.1), but not the CV condition ((cv(ISI.) * 1). G: Example edge-case of the CV metric: network fulfills the CV condition, but not the auto-covariance condition. H: Left: Example plausible candidate nule simulated in two networks with different input rates. Right: pre-post pairing protocol on the 4 co-active rules.
(Methods. 2.4). F: Example edge-case of the auto-
rule was parameterized as a polynomial (Eqn 6 [S),, with a total of 24 parameters 0 (6 parameters per connection type). We applied fSBI to this 24 24-dimensional search space, to identify and retain the rules that established biologically plausible dynamics in SNNs. To ensure the inferred rules robustly enforced plausible dynamics in SNNs, any simulation was assigned a random input rate and connectivity matrix.
We started ISBI with uniform priors for all plasticity parameters: TXy, TXY ~ U(10, 100]ms, and axY, ,xy,, 7xy, AXY ~ 2, The resulting rules formed the initial prior To. We then sampled multiple plasticity rules 0 ~ To. simulated SNNs with these plasticity rules, and saved their spike- trains and weight evolutions for offline analysis of the network metrics x (see Section 2.4). For the first round, we trained Neural Posterior Estimation (NPE) to produce posteriors conditioned only on a subset of metrics the excitatory and inhibitory population firing rates (e)xe rinh). Next, we sampled plasticity rules from this posterior corresponding to networks with excitatory and inhibitory rates between and 50 Hz, i.e., 0' p(0')rexc E [1, 50)Hz, Tinh E [1, 50) H2). These rules were used for the next round of inference, which we conditioned on the metrics associated with stable weight dynamics (fuea Wersep (wxy)). in addition to the population activity rates. We thus made the posterior from the previous iteration, p(0) Texe [1, 50]H2, Finh E 50) into the prior nı (0) of the current iteration. We progressively applied all above mentioned metrics (defined in Section 2.4 in the order of the least to the most restrictive metric, to maximize sample efficiency (Fig 23. C).
pre pre
We quantified the behaviour of the network with a range of metrics [3] and noticed that some networks satisfied some conditions, but not others which should discern similar qualities. For example, some networks passed the auto-covariance criterion (())., < 0.1) but not the CV criterion ((cv(ISI,)) > 0.7), by allowing small population of neurons with unrealistically high firing rates to "cheat" the metric (Fig. 2 F). Conversely, other networks passed the CV condition but failed the auto-covariance one, e.g., with a bimodal inter-spike interval (ISI) distribution (Fig, 2G). To

---

filter out these edge cases, we used several metrics for each desired aspect of the network dynamics (Section 2.4).
We found that ISBI successfully identified relevant parts of the search space: the 10,000 samples from the uniform prior To contained no plausible rules (i.e., no rules producing networks satisfying all criteria). However, by sequentially adding constraints we arrived at posterior (T(0) that, at the end of the fourth search round, led to more than 50% suitable rules (Fig. Interestingly, the dimensionality of the posterior, i.e., of the "suitable rule space", shrunk as we sequentially refined the criteria for plausible network dynamics (Fig. (D), suggesting that fSBI captured specific relationships between the plasticity parameters. On the other hand, the final posterior remained relatively high dimensional, comprising a variety of different rules, with no apparent posterior collapse onto a single rule. The plausible rules sampled from this final posterior showed stable activity and weight dynamics, regardless of their input rates (Fig--1)
Interdependencies between plasticity parameters in the fSBI inferred rules: Next, we examined interdependencies between ISBI-inferred parameters as a possible explanation for the reduced effective dimensionality of the parameter space (Fig. 2D). We found that the correlation was overall weak between the parameters of the plausible rules (Supp. Fig. 6.2.3. This was not surprising, we compared a range of rules that establish a wide diversity of SNN dynamics, such that individual rules can be expected to have different parameter dnterdependencies. We thus narrowed our investigation to rules that produced excitatory firing rates of 10Hz (as opposed to range of rates between I and 50Hz, Fig. BA) The corresponding correlation matrix revealed a ane-grained structure for parameters within the same rule (block diagonal structure), where we observed mostly negative correlations, especially between the non-Hehbian parameters within each rule, (xx and fxx.. Additionally, we found both positive and negative correlations between any given connection types (off- diagonal structure, Fig. A), suggesting that plasticity mechanisms must be co-tuned to function properly.
ISBI rules improve upon mean-field predictions: We also compared the ISBI-inferred rules to mean-field predictions. Mean-field analysis links the plasticity rule parameters the average expected firing rate (Section 2.3] Fig. BB), thus allowing us to obtain theoretically viable rules. Samples from the initial prior did not show any similarity to theoretical predictions when projected along the mean-field axes (Fig. B3C, left). As rules were filtered by progressively more constraints, a similar structure to the mean-field predictions emerged, albeit confined to firing rates below 10Hz, possibly because of synchronisation effects for higher firing rates (Fig. 33C for EE; similar conclusions for the other three synapse types in Supp. 6.2.. Interestingly, the set of rules predicted by mean field theory was not superset of those selected by ISBI: Some rules were exclusively predicted by one but not the other approach (Fig. E). Notably, many rules predicted by mean-field, but not by ISBI, did not result in plausible dynamics when tested numerically (Fig. 3D, middle). In contrast, several rules predicted by fSBI but rejected by mean-field proved numerically viable (Fig. B middle). Finally, the EE rules predicted by both mean-field and fSBI were plausible (Fig. (Fig.3., bottom). We noted that ISBI rules lead to plausible network dynamics both when simulated with only the EE parameters (keeping the weights of the other connection types fixed, Fig. BE. bottom), and also when using the rule for all connection types (Fig. 3 middle), thus confirming the success of ISBI.
Comparing rules under pre-post protocols: We visualized the fSBI rules under the pre- prep pairing protocol widely used experimentally and theoretically to uniquely identify STDP rules We found that among the various rules simulated, there was no apparent link between how rules appeared in the pre-post protocol, and their ability to establish plausible dynamics in SNNs. Some rules appeared near-identical under the pre-post visualization and yet resulted in vastly different network dynamics (Fig. Conversely, rules with visibly differing pre-post protocols established similar network dynamies (Fig. 2H and Fig. 3). Taken together, these results question the relevance of such classical pre-post protocols in elucidating and uniquely determining the plasticity rules at play in the brain, particularly when they are performed without additional disambiguating experiments.
3.2
ISBI infers plausible rules from an MLP search space
We wanted to expand our search space to include local and semi-local variables that have been theoretically or experimentally linked to synaptic changes 244, such as the efficacy of the synapse the post-synaptic membrane potential [II]. the spike-history of both the pre- and post-synaptic neurons 222, as well as the activity of neighboring excitatory and inhibitory synapses (co-dependent
44
plasticity [10). Since flexibly combining these variables with polynomial would result in very
7

---

A
EE rules predicted by mean-field, plotted (0 Fea. 10lb)
D
Mean-field posterior (EE)
E
(SBI posterior (15))
Tar
high
distance to its
high
distance to
MF posterior
Taeat
Ber
8ac
Tpost
low
low
cov 0/1
Full rule
-0.2
*
0.2
neurons
0.0
0.2
Tpm
10
Tpost
Aie twe
(s)
100 ms
100 ms
0.2
0.2
Toost
-0.
0.2
neurons
Only EE
0.22
0.2
10
A2 (S)
Spr
T 0 TToott
Tpoer B
Tpore B
100 ms
100 ms
AC (s
B Mean-field predictions
All rules
Stable activity
Stable weights
Plausible
0.4
30
0.4
30
0.4
30
0.4
30
0.4
30
Aze
15
Ace
15
15
15
15,
-0.4
0.4
-04
0.4
0
-0.4% -Mee
PEE
Pee
-aee
PEE
Per
Figure 3: Polynomial JSBI rules compared to rules predicted by mean-field.
A: Pearson correlation matrix of plasticity rules sampled from the final ISBI posterior (Ta)), con- ditioned on the excitatory firing rate Texe = 10Hz. B:
alongside the relevant parameter combinations (eq. 7 C: Rules sampled from fSBI posteriors, with an increasing number of conditions, along the same axes as in B. D: Same set of rules as in B. but shown along of the 6 plasticity parameters (a and 3), color coded by the mean Euclidean distance between each rule and the 10 closest -SBI-inferred EE rules. Middle (resp. bottom), network dynamics with a rule among the furthest (resp. closest) from ISBI samples, and the corresponding pre-post protocol. E: Rules sampled from the final fSBI posterior T3, shown along the same axes as D. color coded by the mean Euclidean distance between each rule and the 10 closest mean-field plasticity rules. Below, network dynamics with one of the rules furthest from mean- field, and the corresponding pre-post protocol, either the full rule (EE, EI, IE, II, middle) only the EE part of the rule (bottom). Weight traces in Supp. 6.6
high-dimensional, hard-to hard-to-navee tavviggte search spaces, we switched to multi-layer perceptrons (MLP). The MLP- rules we designed used the aforementioned synaptic variables as inputs at the time of a spike, and produced a weight change for a given synapse (Fig. 4B), thus representing the plasticity rule. Crucially, we only trained the final layer of that MLP, thus allowing for potentially highly non-linear interactions between the plasticity variables while retaining a relatively low dimensional (searchable) parameter space. We verified that such MLP- au.-parrmeterized rules can approximate established plasticity rules on supervised tasks (see Supp. 6.6 and then applied ISBI to inferring co-active EE and IE rules (Fig. [4A B). Like for the polynomial space, we found that iSBI produced progressively more constrained posteriors, such that the final ISBI posterior generated over 50% plausible rules (Fig. C). Furthermore, fSBI reduced the effective dimensionality of the parameter space, i.e., increased correlations between the parameters (Supp/6.7). like what we had observed for the polynomial expression of rules. Interestingly, plausible rules from the final fSBI posterior produce stable activity and weight dynamics, but with unique pre-post protocol profiles that are markedly different from that of experimentally observed rules, as well as the rules discovered by the polynomial method above (Fig. AD. F,, 6.7p These rules represent potentially complex and non-linear interactions between Hebbian, local and semi-local factors, but nevertheless robustly establish plausible network dynamics.
Generalization capabilities of meta-learned rules: Finally, we verified that both the polynomial and the MLP rules obtained with fSBI generalized to different network sizes, sparsity, weight initializations, and ratios of
although few metrics were sometimes outside the predefined ranges (Fig. 54) We then tested the "plausible" rules on 20-minute- minutteleoon simulations (versus 2min in Fig.2 4.. Approx. 25% of the polynomial rules still met the conditions on all 15 metrics after 20 minutes (Fig. 5B. Most other rules were disqualified by our (WE) and (we) conditions, which select long-term stable weights, and are thus sensitive to simulation duration. Correspondingly, the II and IE plasticity parameters appeared more refined compared to the starting set (Fig. 5D). Overall, this
8

---

A
pre
fraction rules sampled from
(W)
Sport
Whl
Waz
W post
10"*
stable activity
Exc
Inh
Wore
ndw
10
stable weights
10-3
A.l. dynamics
<10--
plausible
Text
R1g
72
n3
Ah1
Mla
D
E
10Hk)
Example plausible from itg
Text
SHz
Text
15Hz
high
Wine
Inh EXC
activity
W 4
covie,. 0,)
(H2)
20
20
low activity
high activity
0.4
Wee
0.4
Wee
0
Awe
1.2
1.2
0.8
0.8
low
60
-0.1
0.1
activity
(s)
(s)
At
Were
Wpest
Wore
Wacst
Figure 4: Posterior of plausible plasticity rules fSB.. MLP search-space.
A: Recurrent spiking network receiving Poisson inputs at random rate Texts with recurrent EE and IE plastic connections parameterized as multi-layer-perceptrons (MLP). B: Weight changes via forward pass through MLP that depends on pre- and post-synaptic traces, current synaptic weight, smoothed membrane potential and two codependency terms at every spike (Methods. (22)). C: Fraction of rules sampled from each intermediate fSBI posterior that obey the corresponding network metric condition. D: Left: Example plausible candidate rule simulated in two networks with different input rates. Right: pre-post pairing protocol on the EE and IE rules. Note that these rules have different outcomes on the pre-post protocol depending on the network activity (Supp 6.4). E: Pearson correlation matrix of the plasticity rules sampled from the final fSBI posterior, conditioned on the excitatory firing rate being 10Hz.
suggested that the 2min task had properly constrained the EE and rules, but was too short fully constrain the II and IE rules. Similarly, 35% of the 2min-plausibe MLP rules met all conditions in the 20min task. Interestingly, dimensionality reduction of the successful plasticity parameters did not reveal sub-structures, suggesting that the 20min task did not dramatically refine the set of filtered rules (Fig. 5C).
4 Discussion and Outlook
Plasticity mechanisms are arduous to probe in vivo and theoretically challenging to describe due to the complex interplay between neural activity and weight dynamics. Here, we introduce fSBI, a tool numerically infer families of plasticity rules while enforeing strong plausibility constraints SNN dynamics. We show that fSBI can discover generally plausible plasticity rules in large SNNs, with rules parameterized either as polynomials MLPs.
(SBI has several advantages compared to previous approaches: analytical methods require various and sometimes strong assumptions; local optimization methods only provide single solutions; and brute-force approaches do not scale (in this study, randomly drawing from the full space of plasticity rules would have required simulating ~00M rules, compared 200k, Fig. 2 4C). However, these approaches are not mutually exclusive: analytical frameworks allow us to understand the sets of rules inferred by (SBI and produce meaningful experimental predictions. Furthermore, the ISBI distributions of plausible rules are promising starting point for further investigation into specific network computations, potentially suited to local optimization approaches.
In contrast with previous in silico studies of plasticity that produce single predictions, fSBI is systematic approach to make predictions based on many (if not all) candidate plasticity rules within search space. For instance, we showed how an isolated EE rule may suggest a Hebbian mechanism, but the (SBl-discovered set contains equally valid anti-Hebbian rules that can establish near identical

---

4
Polynomial rules
MLP rules
Sparsity xo.5
respecting
0.8
Mosures x2
fraction of
fraction
ratio 3:1
E:1 ratio 5:1
10.5
Feer
Fre
activity
Weow
(Wed
(we))
(m9
weights
A.
all
activiy
Wereo
(Wedl
weights
Wext
Polynomial
MLP
fraction of rules
respecting
2.5
SSWE
yes
after 20min
0.
PCA,
stable
0.0
-50
Den
actolt
Worep
(Wee)
(we)
(We)
(or))
weights
cxtS0
A.L
all
0.0
2.5
-50
0
50
no
PCA:
t-
SNE
yes
stable after 20min
Ba
-2
-2
no
0
-2
Oze
der
dg
Figure 5: Robustness of polynomial and MLP rules.
A: 1k polynomial (left) and 1k MLP (right) rules that fulfilled all plausibility conditions on the min task (Fig (4 were simulated the same task, but with different probability of connection (5%), network size (10k, 20k), E:1 ratios or initial weights (wit = "El
init
wil = 0.05.
TO
or wint WEl W " 0.15). All simulations had random input strength and initial connectivity. "Activity", "weights" and "A.I" denote the intersection of all metrics in that category. B: 10k "plausible" polynomial and 10k MLP rules were simulated on 20-minute extended version of the 2-minute background task ( Fraction of the rules that fulfill some plausibility conditions on the 20-minute task. C: First two features of PCA (left), t-EE (right) applied to the MLP rules in B. Rules fulfilling all plausibility conditions on the 20-minute task are in yellow. D: -ebbaan parameters of the polynomial rules from B, with the same color code as C.
network activity (Figs. 2 E)). suggesting that studies of single rules or individual connection types may be grossly under-constrained.
More generally, our study challenges the widely used pre-post pairing protocols 6 44] when used in isolation. Indeed, rules that look near identical under such pre-post protocols may have widely different network-level effects (Fig. BD, middle and bottom), validating, in hindsight, the need for finely-tuned orchestration in previous studies [8]. Conversely, rules that appear different under observation with a pre-post protocol (Fig. 2 and Fig. 3) may lead to similar network dynamics. In addition, the results above were obtained in noise-free in silico experiments, suggesting that the connection between empirical data and plasticity rules may be even more complex. The interpretability of the pre-post pairing protocols is further compromised in the MLP-based rules, since these rules include factors linked to plasticity that are not explicitly constrained by such low- dimensional protocols, leading widely different outcomes depending on the synaptic and neuronal state (Fig. 4D). A Additionally, we have not touched on the effects of neuromodulation in our study, something that currently out of scope but will become the focus of future studies.
Our approach naturally comes with limitations and idiosyncrasies. First, we cannot prove the long- term stability of the rules, compute requirements. On longer timescales, slower homeostatic mechanisms may play a part in synaptic dynamics 455 46). Another potential limitation of fSBI is the curse of dimensionality, inherent to most SBI approaches [36], although some recent improvements have been proposed 47-50). Finally, further analysis is needed forge robust and meaningful predictions from the commonalities of the large set of inferred rules.
limited by the numerical approach and associated demanding
In summary, meta-learn plasticity rules with inference- anference-based method, ISBI. This method can be applied to large spiking networks with flexibly parameterized plasticity rules and narrow down the set of promising plasticity mechanisms.
10

---

5 Acknowledgments
We thank Chaitanya Chintaluri, Everton Agnes, Nicoleta Condruz, Douglas Feitosa Tomé, Michael Deistler and Jan Boelts for helpful discussions and feedback on the manuscript. This work was funded by the European Research Council (ERC consolidator grant SYNAPSEEK), the German Research Foundation (DFG; Germany's s Excellence Strategy MLCoE EXC number 2064/1 PN 390727645), the German Federal Ministry of Education and Research (BMBF: Tübingen AI Center, FKZ: 01IS1 18039A,, the Human Frontier in Science Program (RGY0076/201)) and the FENS- Kavlii Network of Excellence scientific exchange program. This research was supported by the Scientific Service Units (SSU) of IST Austria through resources provided by Scientific Computing (SciComp).
References
[1] Tim VP Bliss and Terje Lømo. Long-lasting potentiation of synaptic transmission in the dentate area of the anaesthetized rabbit following stimulation of the perforant path. The Journal of physiology, 232(2)3331-35,, 1973.
[2] Larry F Abbott and Sacha B Nelson. Synaptic plasticity: taming the beast. Nature neuroscience, 3(1)):1188-118, 2000.
[3] Jeffrey C Magee and Christine Grienberger. Synaptic plasticity forms and functions. Annual review of neuroscience, 43:95-117, 2020.
[4] Ami Citri and Robert C. Malenka. Synaptic plasticity: Multiple forms, functions, and mecha- nisms. Neuropyychpharrmcollgg, 33:18-41, 2008.
[5] Amanda R. McFarlan, Christina Y. C. Chou, Airi Watanabe, Hannah Owens Nicole Cherepacha, Maria Haddad, and Jesper Sjöström. The plasticitome of cortical interneurons. Nature Reviews Neuroscience, 24, 2022.
[6] Wulfram Gerstner, Richard Kempter, J Leo Van Hemmen, and Hermann Wagner. A neuronal learning rule for sub-millisecond temporal coding. Nature, 383(6595))76-788 1996.
[7] Tim P Vogels, Henning Sprekeler, Friedemann Zenke, Claudia Clopath, and Wulfram Gerst- ner. Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks. Science, 1546 1573, 2011.
[8] Friedemann Zenke, Everton J Agnes, and Wulfram Gerstner. Diverse synaptic plasticity mechanisms orchestrated to form and retrieve memories in spiking neural networks. Nature communications, 6(1):1-13, 2015.
[9] Ashok Litwin-Kumar and Brent Doiron. Formation and maintenance of neuronal assemblies through synaptic plasticity. Nature Communications, 2014.
[10] Everton J Agnes and Tim P Vogels. Codependent excitatory and inhibitory plasticity accounts for quick, stable and 2022.
memories in biological networks. BioRxiv, page 437962,
[11] Claudia Clopath, Lars Büsing, Eleni Vasilaki, and Wulfram Gerstner. Connectivity reflects coding: a model of voltage-based stdp with homeostasis. Nature Neuroscience, 13:344-352, 2010.
[12] Sukbin Lim, Jillian L. McKee, Yali Woloszyn, Luke Amit, David J. Freedman, David L. Sheinherg, and Nicolas Brunel. Inferring learning rules from distribution of firing rates in cortical neurons. Nature neuroscience, 18:1804 8:1804-1811 2015.
[13] Yoshua Bengio, Samy Bengio, and Jocelyn Cloutier. Learning a synaptic learning rule. Citeseer, 1990.
[14] Keren Gu, Sam Greydanus, Luke Metz, Niru Maheswaranathan, and Jascha Sohl- Sohh-Dickstein. Meta-learning biologically plausible semi-supervised update rules. bioRxiv, 2019.

---

[15] Basile Confavreux, Friedemann Zenke, Everton J Agnes, Timothy Lillicrap, and Tim P Vogels. meta-learning approach (re) discover plasticity rules that carve desired function into neural network. Advances in Neural Information Processing Systems 34 (NeurIPS), 2020.
[16] Luke Metz, Niru Maheswaranathan, Brian Cheung. and Jascha Sohl- Sohl-Dicksteim. Learning unsupervised learning rules. arXiv preprint arXiv:1804.0022 2018.
[17] Jakob Jordan, Maximilian Schmidt, Walter Senn, and Mihai A Petrovici. Evolving interpretable plasticity for spiking networks. eLife, 10:e66273, 2021.
[18] Jack Lindsey and Ashok Sttwin-Kumar, Learning to learn with feedback and local plasticity. Advances in Neural Information Processing Systems 34 (NeurIPS), 2020.
[19] Danil Tyulmankov, Guangyu Robert Yang, and LF Abbott. Meta-learning synaptic plasticity and memory addressing for continual familiarity detection. Neuron, 110(3):344 557, 2022.
[20] Navid Shervani-Tabar and Robert Rosenbaum. Meta-learning biologically plausible plasticity rules with random feedback pathways. Nature Communications, 14(1):1805, 2023.
[21] Wulfram Gerstner and Werner M Kistler. Mathematical formulations of hebbian learning. Biological cybernetics, 8755):004-415, 2002.
[22] Jean-Pascal Pfister and Wulfram Gerstner. Triplets of spikes in a model of spike timing- dependent plasticity. Journal of Neuroscience, 26(38):9673 9682, 2006.
[23] Wulfram Gerstner, Werner M Kistler, Richard Naud, and Liam Paninski, Neuronal dynamics: From single neurons networks and models of cognition. Cambridge University Press, 2014.
[24] Peter Dayan and Laurence F Abbott. Theoretical neuroscience: computational and mathematical modeling of neural systems. MIT press, 2005.
[25] Nicolas Brunel. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of computational neuroscience, 8:183-208, 208, 2000.
[26] Tim P Vogels, Kanaka Rajan, and Larry Abbott. Neural network dynamics. Annu. Rev. Neurosci., 28:357-376, 2005.
[27] Alfonso Renart, Jaime De La Rocha, Peter Bartho, Liad Hollender, Néstor Parga, Alex Reyes, and Kenneth D Harris. The asynchronous state in cortical circuits. science, 32755665) 2010.
[28] A. Sanzeni, M. H. Histed, and N. Brunel. Emergence of irregular activity in networks of strongly coupled conductance-ban based neurons. Physical Review X, 12((1:01 1004, 2022.
[29] Michael N Shadlen and William T Newsome, Noise, neural codes and cortical organization. Current opinion in neurobiology, 4(4):569-579, 1994.
[30] Zachary Mainen and Terrence Sejnowski. Reliability of spike timing in neocortical neurons. Science, 26065166 1995.
[31] Sen Song, Per Jesper Sjöström, Markus Reigl, Sacha Nelson, and Dmitri B Chklovskii, Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS biology, 3(3)e66, 2005.
[32] Michael Okun and Ilan Lampl. Instantancous correlation of excitation and inhibition during ongoing and sensory-evoked activities. Nature neuroscience, 11(5):535-537, 2008
[33] Jonathan W Pillow, Jonathon Shlens. Liam Paninski, Alexander Sher, Alan M Litke, EJ Chichilnisky, and Eero P Simoncelli. Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207):995-999) 2008
[34] Marlene R Cohen and Adam Kohn. Measuring and interpreting neuronal correlations. Nature neuroscience, 14(7):81 111899 2011.
12

---

[35] George Papamakarios and lain Murray. Fast e-free inference of simulation models with Bayesian conditional density estimation. In Advances in Neural Information Processing Systems 29, pages 1028-1036. Curran Associates, Inc., 2016.
[36] Kyle Cranmer, Johann Brehmer, and Gilles Louppe. The frontier of simulation-based inference. Proceedings of the National Academy of Sciences, 1 117/48):30055-30022 2020.
[37] Pedro Gonçalves, Jan-Matthis Lueckmann, Michael Deistler, Marcel Nonnenmacher, Kaan Öcal, Giacomo Bassetto, Chaitanya Chintaluri, William F Podlaski, Sara A Haddad, Tim P Vogels, et al. Training deep neural density estimators to identify mechanistic models of neural dynamics. Elife, 9:c56261, 2020.
[38] Jan-Matthis Lueckmann, Pedro Goncalves, Giacomo Bassctto, Kaan Ócal, Marcel Nonnen- macher, and Jakob H Macke. Flexible statistical inference for mechanistic models of neural dynamics. Advances in neural information processing systems, 30, 2017.
[39] David Greenberg, Marcel Nonnenmacher, and Jakob Macke. Automatic posterior transformation for likelihood-free inference. In International Conference on Machine Learming, pages 2404- 2414. PMLR, 2019.
[40] Michael Deistler, Jakob Macke, and Pedro Gonçalves. Energy-efficient network activity from disparate circuit parameters. Proceedings of the National Academy of Sciences, 119(44): e2207632119, 2022.
[41] Michael Deistler, Pedro J. Goncalves, and Jakob H. Macke. Truncated proposals for scalable and hassle-free simulation- based inference. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, Advances in Neural Information Processing Systems, 2022.
[42] Conor Durkan, George Papamakarios, and lain Murray. Sequential neural methods for likelihood-free inference. Bayesian Deep Learning Workshop at Neural Information Processing Systems, 2018
[43] Henry Markram, Joachim Lübke, Michael Frotscher, and Bert Sakmann. Regulation of synaptic efficacy by coincidence of postsynaptic aps and epsps. Science, 275(5297) 1997.
[44] Guo-qiang Bi and Mu-ming Poo. Synaptic modifications in cultured hippocampal neurons: de- pendence on spike timing, synaptic strength, and postsynaptic cell type. Journal of neuroscience, 18024):10464-1014) 1998.
[45] Keith B. Hengen, Alejandro Torrado Pacheco, James N. McGregor, Stephen D. Van Hooser, and Gina G. Turrigiano. Neuronal firing rate homeostasis is inhibited by sleep and promoted by wake. Cell, 165(1):180-191, 2016. ISSN 0092-8674.
[46] Gina G. Turrigiano. The self self-tuning neuron: Synaptic scaling of excitatory synapses. Cell. 135 (3):422-434 2008. ISSN 0092-8674 8674.
[47] Poornima Ramesh, Jan-Matthis Lueckmann, Jan Boelts, Álvaro Tejero- Cantero, David Greenberg, Pedro J. Goncalves, and Jakob H. Macke. GATSBI: Generative adversarial training for simulation-hased inference. In International Conference on Learning Representations, 2022.
[48] Lorenzo Pacchiardi and Ritabrata Dutta. Likelihood-free inference with generative neural networks via scoring rule minimization. arXiv preprint arXiv:2205. 15784, 2022.
[49] Victor M.-H. Ong, David J. Nott, Minh-Ngoc Tran, Scott A. Sisson, and Christopher C. Drovandi. Likelihood-free inference high dimensions with synthetic likelihood. Computational Statistics & Data Analysis, 128:271 18::271-29 1911, 2018. ISSN 0167-9473.
[50] Athanasios Kousathanas, Christoph Leuenberger, Jonas Helfer, Mathieu Quinodoz, Matthieu Foll, and Daniel Wegmann. Likelihood-free inference in high-dimensional models, 2015.
[51] Friedemann Zenke and Wulfram Gerstner. Limits to high-speed simulations of spiking neural networks using seneral-purposs computers. Frontiers in neuroinformatics, 8:76, 2014.
13

---

[52] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Coumaaeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fer- nández del Río, Mark Wiehe, Pearu Peterson, Pierre Gérard-Marchant Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, and Travis E. Oliphant. Array programming with NumPy. Nature, 585(7825):357-362, September 2020.
[53] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournpeea, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 222882-28830, 2011.
[54] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C.J Carey, Ilhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261-272, 2020.
[55] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Z. Yang, Zach De Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-performance deep learning library. CoRR, abs/1912.01 703, 2019.
[56] Alvaro Tejero-Cantero, Jan Boelts, Michael Deistler, Jan-Matthis Lueckmann, Conor Durkan, Pedro J. Gonçalves, David S. Greenberg, and Jakob H. Macke. sbi: A toolkit for simulation- based inference. Journal of Open Source Software, 5(52):2505, 2020.
[57] D. Hunter. Matplotlib: 2d graphics environment. Computing Science Engineering, (3):90-955 90000-95,220 2007.
[58] Nikolaus Hansen. The CMA evolution strategy: A tutorial. arXiv preprint, 1604.00772, 2016.
14


Reviews:
[{"review_id": "ER7nXhGOoD", "reviewer": "Reviewer_vJPD", "summary": "How neurons update their synaptic strengths is an important and active area of research in neuroscience, with potential applications in ML. Much work has been done using analytically tractable mean-field models, in large part because the search space for possible rules is intractably large.  Here, the authors introduce an approach that cuts the search space down significantly.  While simulation-based inference has been used to infer parameter distributions that lead to network dynamics, the authors add constraints on plausible network dynamics to further reduce the search space to a manageable size.  \n\nThey find that plasticity rules predicted by the mean field model they derive do not always result in plausible dynamics in the full network.  Moreover, pre-post diagrams that are often used in STDP analyses were found to be of limited use: rules with very similar diagrams could drastically diverge in terms of network dynamics, while rules with very similar network dynamics could result in very different pre-post diagrams.", "questions": "", "limitations": "Yes", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Overall, this is a good study that contributes to the field. Brute force strategies are inherently problematic; the authors' filtering approach provides a helpful middle ground between approaches based on more tractable methods and full brute-force grid searches.  The comparison of the rules between the full and mean-field networks, as well as the findings of the pre-post diagrams, add nuance to the study of plasticity in computational and experimental settings.", "weaknesses": "Major:\n\nThe filter constraints are hard boundaries that are not always obviously well-founded.  For example, it isn't clear why firing rates above 50 Hz are considered implausible when population gamma oscillations typically go up to ~80 Hz and neurons have been found to fire up to ~1000 Hz. Why the bounds for $<w_{EX}>$ and $<w_{IX}>$ are an order of magnitude apart is unclear, nor is there physiological comparison (weights are in arbitrary units). Other bound decisions are similarly left ambiguous.  \n\nIf the authors believe there are good reasons to have chosen the specific values presented, they should explain and cite the relevant work.  Otherwise, the paper should be rewritten to soften the claims.  In particular, the authors could instead choose to indicate that others might choose somewhat different bounds to be more or less inclusive, but that this is a first attempt and that the main conclusions are left unaffected.  The \"plausible\" network dynamics claims should thus be de-emphasized in that case.\n\nMinor: \n\nThe MLP plasticity rule was introduced in a manner that was difficult to understand.  However, I found my questions answered when the results were introduced.  In particular, lines 252-264 should either be moved to line 82, when the rule is introduced (or integrated into that section), or vice versa.\n\nThe color bars in Fig 3D-E should have scales; if the lower bound is not very near zero, some of the associated claims may need to be softened.\n\nLine 84: $C_i^I$ should be $C_i^{\\mbox{inh}}$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the constructive feedback, and for noting the potential of the approach. Please find a detailed response to the points that are not addressed in the common rebuttal below. \n\t\n### Choice of metrics and their plausible ranges: \n\nWe addressed most of this point in the common rebuttal above. Concerning the definition and interpretation of the weights in our model, weights affect the membrane potential via the driving force (membrane potential minus reversal potential, conductance-based model), which is different between excitatory and inhibitory synapses. This, in addition to the difference in the numbers of excitatory and inhibitory neurons, leads to requirements of larger inhibitory weights to achieve a balanced state, or close to ( Sanzeni 2023, Brunel 2000). In addition, synaptic weights are unitless because they are in units of leak conductance (10nS), but still comparable to experimental data via their potentials (PSPs), as mentioned in the common rebuttal.\n\nThe lower bound of 1Hz was chosen mainly for numerical reasons: allowing us to collect enough spikes in a given 2min simulation and thus meaningfully assess rules with spike-triggered updates.\n\nIn addition, we chose the ranges for the less widespread metrics, such as $\\langle \\sigma \\rangle_i, \\langle S \\rangle_{i, f}, \\langle Fano \\rangle_i, \\langle \\rho \\rangle_{i, t}, \\langle Fano \\rangle_t, w_{creep}, f_{w_{blow}}$, by benchmarking the respective metrics on example networks such as Poisson spike trains at various rates, networks from classical studies in the field (Vogels et al. 2011, Zenke et al. 2015, Litwin-Kumar & Doiron 2014) and also implausible networks —e.g., synchronous-regular regime.\n\n$f_{w_{blow}}$ was introduced to reject biologically unrealistic rules that led more than 10% of the weights of any synapse type to 0 or to the maximum weight $w_{max}$.\n\n$w_{creep}$ was designed to reject weights that changed unrealistically fast, excluding mean relative changes of more than 5% over the final 10s of simulation.\n\nWe will make these motivations clear also in the manuscript.\n\n### Minor comments:\nWe will rephrase how the MLP rules are introduced in line 82 according to feedback.\n\nColorbar Fig 3D,E. This was an oversight, apologies. Black corresponds to large distance and yellow to 0. We will add the legend in the final version.\n\nWe will correct the typo line 84.\n"}, {"review_id": "DbqaDmIQJb", "reviewer": "Reviewer_5no9", "summary": "The manuscript introduces a novel way of sampling plasticity rules that obey certain plausibility constraints. Specifically, the authors focus on networks of excitatory and inhibitory spiking neurons and use a simulation-based inference (SBI) technique to filter plasticity rules based on the behavior of the networks in which they are implemented. They demonstrate the effectiveness of the technique, and in particular find that standard neuroscience protocols for examining synaptic plasticity are insufficient for identifying plasticity rules.", "questions": "1. Do we need to worry about any biases in the sampling obtained using fSBI?  If there are such biases, would they depend markedly on details such as the order in which the constraints are imposed?\n2. How were the details of the simulations mentioned above (like size of networks) chosen? Were multiple choices explored and the choice did not matter? Were some of the choices informed by measurements in real brains? If so, which areas and which animals -- and can we expect the results to generalize?\n3. On line 82 it is written that *\"For the MLP plasticity rule, only EE and IE synapses undergo spike-triggered updates\"*. Why? Is this an arbitrary choice, or is there something deeper about it?\n4. The comment on lines 268–270 suggests that many of the MLP rules are not, in fact, realistic. Should we then not try to either simplify the neural nets or add more constraints to restrict ourselves to realistic rules?\n\nMinor comments:\n* on line 86, what are the superscripts (1), (2)? are they the same as *pre*, *post*?\n* on line 195, *\"the order of the least to the most restrictive metric\"*: how did you know which metrics were more restrictive?\n* Figure 3, panel A: what do the colors for the axis labels represent?\n* Figure 3, panels D, E: which side of the color bar is for small distances and which is for big?\n* on lines 239-240: *\"fSBI rules lead to plausible network dynamics both when simulated with only EE parameters [...] and also when using the rule for all connection types\"*: why would we expect / want the dynamics to be stable when the plasticity rule applies only to some of the connections in the network?\n* line 257: should mention the activation function(s) used for the MLP", "limitations": "The authors have adequately discussed limitations of their work.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "*Originality:*\nThis is a nice extension of simulation-based inference that significantly reduces the size of the search space by enforcing constraints one by one instead of all at the same time.\n\n*Quality:*\nThe work seems carefully done, with a significant number of simulations to support their conclusions.\n\n*Clarity:*\nThe presentation is clear and the paper is well-written.\n\n*Significance:*\nUnderstanding the plasticity rules used by neural circuits is of utmost importance in neuroscience. This paper addresses this question in an interesting way, and the authors show shortcomings with standard neuroscientific approaches.", "weaknesses": "There are a number of choices related to the simulations that seem insufficiently discussed: e.g., the ratio of excitatory to inhibitory neurons in the networks that were simulated; the sizes of these networks; the statistics of the inputs to the networks; the initial weight distribution; the duration of the simulations; the various numerical constants used in the metrics from section 2.4.\n\nAlso, while the authors point out that standard protocols from neuroscience do not uniquely pin down bio-plausible plasticity rules, it would be good to at least speculate on possible approaches that could solve this problem.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thoughtful review, please find below our attempt to address the points raised that are not in the common rebuttal.\n\n### 1/ Biases in fSBI:\nThe reviewer's concern about biases of the method and the influence of the filtering order is valid, as fSBI may suffer from the same biases as other simulation-based inference (SBI) algorithms: poorer posterior estimation due to limited training samples, curse of dimensionality, etc, which we discuss in the main paper.\n\nBiases may also be introduced by the fact that fSBI does not target the true Bayesian posterior: for a detailed discussion, we refer the reviewer to supplementary section 6.5. In particular, the constraints are likely not conditionally independent given the parameters, and thus the ordering may affect the posterior estimate. In our case, a less restrictive constraint on the rule space came before a more restrictive one, for sample efficiency. However, the restrictiveness of a rule and the corresponding fSBI posteriors are determined by a multitude of factors, such as the parametrization of the plasticity rule, network size, etc. We expect that biases introduced by constraint ordering, and indeed the ordering itself must vary on a case-by-case basis. Defining a rigorous method for ordering the constraints would require work beyond the scope of this paper.\n\n### 2/ Simulation parameter choices: \nBesides the points made in the common rebuttal, we would like to add that the simulation parameters (network sizes, neuron model parameters, connectivity, initialization, inputs) were chosen to be as consensual as possible in the spiking network community. The parameters broadly emulate mammalian cortical networks and are taken from classical studies on dynamical systems analyses of static recurrent spiking networks (Brunel 2000, Dayan & Abbott 2001 for review, Vogels et al. 2011 and Sanzeni et al. 2023). We have tested the generalisability and robustness of our results on some variations of all of these parameters, with encouraging results. \n\n### 3/ Spike-triggered MLPs: \nThis was mostly a compute choice: running an entire MLP forward at every synapse and every integration timestep is prohibitively compute-intensive. It would be an interesting avenue for future research if a more efficient implementation than ours (that favored simulating many spiking networks in parallel) is possible.\n\n### 4/ “Many of the MLP rules are not realistic”: \nThe MLP rules are more difficult to interpret and we agree that understanding the structure of the plausible rules to make experimental predictions beyond the pre-post protocols is an important avenue for future work. For some specific values of synaptic variables that are not constrained by the protocol, the MLP rules may look like experimentally observed rules. Overall, given how underconstrained these rules are under the pre-post protocol, we do not think that this protocol is a meaningful test for these rules. However, as these rules combine biologically plausible synaptic variables into weight updates that amount to plausible network dynamics, we believe they are a meaningful description of a plasticity rule and have the potential to inform future experiments. We will clarify this in the paper.\n\n### 5/ Speculate on which protocols may better disambiguate plasticity:\nThis is an interesting avenue for follow-up work on our study, and we are currently exploring with some experimental collaborators. However, we have no tangible results to present yet. We believe that indirect data of plasticity rules such as activity dynamics and network behavior, which are more accessible experimentally, will help narrow down the potential plasticity rules at play. \n\n### Minor comments:\n“on line 86, what are the superscripts (1), (2)? are they the same as pre, post?”: Each neuron has two eligibility traces with different timescales, inspired by the fast and slow eligibility traces in the triplet rule (Pfister et al. 2006). So x^(1)_i refers to the eligibility trace with time-constant 1 for neuron i.\n\n“how did you know which metrics were more restrictive?” We computed the fraction of rules that obeyed each metric condition (see Fig 2.C, 4.C) under the posterior estimate at a given inference round. This was taken as a proxy for the condition that was the easiest to fulfill in the next round of inference.\n\n“which side of the color bar is for small distances and which is for big?” This is an oversight from us, apologies. We will put back the scale in the published version: the darker, the higher the distance.\n\n“why would we expect/want the dynamics to be stable when the plasticity rule applies only to some of the connections in the network?” Each synapse type has its own rule in the polynomial search space, despite many previous theoretical studies only considering plasticity at EE synapses, and sometimes at IE synapses as well. We wanted to show that our sets of rules, even though technically sets of 4 co-active rules in a single network, are still plausible when taken in isolation. \n\n“the activation function(s) used for the MLP” is a sigmoid non-linearity. We added that information in the manuscript."}, {"review_id": "YGqoBI0Pks", "reviewer": "Reviewer_jNn6", "summary": "The authors present the filter Simulation-Based Inference (fSBI) method, aimed at automating the discovery of families of rules that establish plausible dynamics within large spiking networks. This study seeks to overcome the challenges presented by human intuition and the overly simplified assumptions that have characterized prior research in synaptic plasticity—a critical element in brain-based learning and memory processes. The fSBI method facilitates the extensive exploration of the potential rule space, identifying sets of rules via metrics that effectively constrain network dynamics to satisfy the intended conditions. Importantly, this approach evaluates one metric at a time, significantly reducing computational complexity compared to simultaneous evaluation of multiple metrics.\n\nOverall, this study provides a principled approach for meta-learning a family of plasticity rules in networks, offering a starting point for deeper analysis of computation and plasticity mechanisms. Additionally, the study highlights that classical experimental predictions often yield contradictory results when considered in isolation, thereby challenging traditional theoretical and experimental protocols for understanding plasticity. Furthermore, the inherent flexibility of this tool lends it the potential to incorporate new neuroscience findings into the metrics and architecture as emerging data flows in.\n", "questions": "o\tAlthough the study acknowledges that slower homeostatic mechanisms aren't considered, how might the inclusion of weight regularization (or homeostatic mechanisms) influence the pool of plausible rules and the interaction between plasticity terms? \n\no\tNeuromodulation is not accounted for in this study, but could the authors briefly discuss (perhaps in Discussion) the feasibility of extending this method to infer rule parameters in recently proposed modulated-eligibility trace-based biologically plausible learning rules for training recurrent spiking neural networks [1-3]?\n\no\tCould the authors provide their intuition/hypotheses on potential reasons why rules appearing similar/different under pre-post protocols can result in different/similar network-level effects?\n\no\tWhat does the resulting weight distribution look like for the plausible candidates? Is it lognormal?\n\no\tReferences: \n\n\t[1] Guillaume Bellec, Franz Scherr, Anand Subramoney, Elias Hajek, Darjan Salaj, Robert Legenstein, and Wolfgang Maass. A solution to the learning dilemma for recurrent networks of spiking neurons. Nature Communications, 11(1):1–15, dec 2020.\n\n\t[2] James M Murray. Local online learning in recurrent networks with random feedback. ELife,8:e43299, 2019.\n\n\t[3] Yuhan Helena Liu, Stephen Smith, Stefan Mihalas, Eric Shea-Brown, and Uygar Sümbül. Cell-type–specific neuromodulation guides synaptic credit assignment in a spiking neural network. Proceedings of the National Academy of Sciences, 118(51), 2021.\n", "limitations": "The authors have acknowledged various limitations: the curse of dimensionality associated with fSBI, limited interpretability of MLP, the use of noise-free simulations, not considering long-term stability measures, among others. These limitations might impact the study's conclusions, including the pool of plausible rules and the nature of interactions between plasticity terms. However, despite these constraints, the study introduces an encouraging framework (fSBI) for inferring plasticity rules that could be enhanced in future research to incorporate additional considerations.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "o\tThe manuscript is very well-written, with informative illustrations.\n\no\tThe proposed fSBI framework exhibits flexibility across architectures and rule parameterization, and can be applied in unsupervised learning contexts.\n\no\tThe viability of the learned rules and their resulting network-level effects is theoretically grounded based on mean-field predictions.\n\no\tAnalysis pertaining to the reduction in effective dimensionality of rule parameters and interdependence of plasticity rule mechanisms is insightful.\n", "weaknesses": "o\tDespite only evaluating one metric at a time, the method may still prove computationally demanding if the number of rule parameters is significant, to accommodate a broader rule basis. The authors recognize this \"curse of dimensionality\", and the Appendix notes simulations took two months.\n\no\tA minor point, but the paper could benefit from more detailed explanation on how specific values for the metric bounds were chosen and how sensitive these choices affect the results (in addition to what's already in 5.2).\n\no\tI could not find information on whether simulations were repeated across different initializations and seeds.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive assessment of our method and its usefulness.\n\n### 1/ Compute requirements:\nWhile we agree that this method is not computationally cheap, this is also true of other meta-learning methods. Notably, our study can be considered as a form of pre-training of plasticity rules, which only needs to be performed once, and then used as a starting point for various investigations into more specific functions.\n\n### 2/ Initialization/seeds:\nEvery rule sampled was simulated once in a network with a new initialization/seed (affecting input strength, connectivity, and the timings of Poisson inputs, denoted as nuisance parameters). Due to the sequential nature of fSBI, robustness to these factors was enforced implicitly, across rounds. The plastic network dynamics shown in Fig 2.H, 3D,E 4D of the main paper were simulated on unseen initialization/seed/input rate, confirming the robustness of the method.\n\n### 3/ Weight regularization/homeostatic mechanism: \nThe MLP rules have a dependence on the current value of the synaptic weights. Some studies propose plasticity rules that are stabilized by considering all the outgoing or incoming weights of a neuron (e.g. Litwin-Kumar 2014, Schulz et al. 2021). It would be an interesting extension for future work to add such terms to the plasticity rules.\n\n### 5/ Neuromodulation and reward-based learning rules:\nThis is an interesting extension to the study, and possible in principle (but out of scope on the short time scale of this review). It either involves defining a new search space and applying fSBI from scratch, or if we consider that reward modulates the learning rule, it may be possible to add a term to the existing search spaces and start from the distributions of rules proposed in this study.\n\n### 6/ Why rules with similar pre-post appear different at the network level:\nThe pre-post protocol only considers a pair of a pre- and a postsynaptic spike, i.e., two spike-triggered weight-updates. For the polynomial search space, this means that the weight change of a protocol is $\\Delta w = \\alpha + \\beta + \\kappa e^{-\\Delta t/\\tau_{post}}$ for (post-then-pre pairs), and $\\Delta w=\\alpha + \\beta + \\gamma e^{-\\Delta t/\\tau_{pre}}$ for pre-then-post pairs. For instance, this protocol does not constrain the exact values of the non-Hebbian terms, $\\alpha$ and $\\beta$, only their sum. So two rules with different mechanisms — e.g., one with no on-pre depression ($\\alpha=0$), and the other with a non-zero $\\alpha$, compensated by on-post depression — may appear identical under the pre-post protocol. However, their underlying parameters, mechanisms, and corresponding network behavior may be very different. In the MLP, there are even more synaptic variables and parameters, which are also unconstrained or partially constrained by the pre-post protocol. This adds even more degeneracy in the rules that could appear identical/similar under the pre-post protocol.\nNote that the analytical considerations above seem to echo experiments: experimental data on these protocols are notoriously difficult to reproduce because the result depends on other experimental variables, e.g., frequency of the spike-pairings, other inputs to the neurons, or history of the connection. \n\n### 7/ Resulting weight distributions:\nWe did not add metrics requiring weight distributions to be lognormal [10], though this would be an interesting future step towards biological plausibility, and we may be able to add it before the paper deadline. The shape of the resulting weight distributions depends on the rule and the synapse type. We have added the weight distribution of the three example rules shown in the main text in the revised Fig.1E. Because we prevent too many weights from blowing up (metrics on stable weight trajectories), we do not expect unnatural, bimodal distributions like in the case of traditional Hebbian learning. We note that when receiving more specific inputs (solving tasks), these distributions may change, so the rule is not the only determinant of the final weight distribution.\n"}, {"review_id": "UpMz6yTorQ", "reviewer": "Reviewer_bG1x", "summary": "The authors propose a novel meta-learning approach based on Bayesian filtering for discovering synaptic plasticity rules that entrain a E-I recurrent spiking neural network to satisfy a set of predefined criteria (e.g. the network firing rates occupy a reasonable regime at steady-state, with appropriate variability statistics, etc.). Using two different parameterizations of their synaptic plasticity rule (a polynomial plasticity rule resembling many different kinds of experimentally observed STDP vs. a multilayer perceptron parameterization) the authors show that they are able to recover many heterogenous rules that match their pre-specified target dynamical regimes, which in some cases align to predictions made by a mean-field model, and in other cases differ in interesting ways.", "questions": "The scaling of the algorithm doesn't seem to be analyzed much beyond one parameterization each for the polynomial and MLP cases which work effectively. Does the MLP method scale well as the number of optimized parameters increases?\n\nHow well do the learned plasticity rules generalize to any of the following: to different network architectures (e.g. multiple layers or different connectivity probabilities), to changes in neuron properties (e.g. excitability or spiking properties), or variations in input statistics?", "limitations": "There are no obvious potential negative societal impacts of the proposed work, and the authors do address many of the limitations of their work (I mention a few that they do not address above).", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper is well written and very clear. Furthermore, the comparisons between the families of plasticity rules derived from mean-field analytic approximations and those derived from fSBI are very interesting--that the authors found plasticity rules that would not have been predicted to be effective under the mean-field approximation is an endorsement of the value of their method, and the fact that several similar-seeming plasticity rules were found to produce markedly different dynamical spiking regimes  has interesting implications for the interpretation of experimental results. The authors' emphasis on multiple different learning rules for different cell types is a welcome improvement over previous methods, and it is a further benefit of their approach that fSBI produces entire families of plasticity rules rather than requiring full commitment to any one particular learning rule (this may allow certain types of predictions across cell types which I elaborate on below).", "weaknesses": "Though the results are technically solid, thorough, and interesting enough to warrant publication in my opinion, I have serious doubts about the long-term value of this conceptual framework overall for several reasons that I think it would be very helpful for the authors to elaborate on.\n\nFirst of all, the learning rules uncovered by fSIB do not actually do anything beyond stabilize the network in a reasonable dynamical regime. It is unclear whether these forms of plasticity can be used for any other interesting functions, such as memory formation or task performance. While this is not an unattainable goal for meta-learning approaches in general (see e.g. Confavreux & Vogels 2020), these alternative approaches use either gradient-based or in some cases evolutionary learning strategies which are likely better suited to searching in high dimensional spaces than sampling-based methods such as fSIB. Without learning to perform ecologically relevant tasks and without comparing to existing alternative methods, we are left wondering whether fSIB is actually useful as an optimization method in this space.\n\nSecond, and relatedly, the flexibility of the plasticity rules learned by fSIB (and most meta-learning alternatives) is limited by the imaginations of the algorithm designers & the scalability of the algorithm. For instance, there are a wide variety of plasticity algorithms that have been proposed for non-trivial task learning that would not be implementable within the parametric space proposed by the authors (e.g. reward-modulated plasticity, as the authors mention; burst-mediated learning as in Payeur et al. 2021; any form of learning involving dendritic compartmentalization as in Urbanczik & Senn 2014; or possibly even forms of phenomenological plasticity based on spiking triplets etc.). The search space is so wide, that all one may do is select a space that accommodates some known set of existing plasticity rules, which makes one wonder whether meta-learning methods for plasticity are in general doing little more than rediscovering algorithms that were already previously identified, but without the theoretical guarantees provided by algorithms motivated by optimization principles.\n\nThis second point would be less relevant if it were possible to generate genuine, testable predictions based on meta-learned plasticity algorithms that go above and beyond those already identified in existing literature. In this study, there appear to be some plasticity rules identified that are not already supported by the mean-field model. However, we are provided with a wide ensemble of potential plasticity parameterizations, and it's unclear what to actually do with this distribution. For instance, could it be used to infer, given a plasticity rule for one type of neuron (e.g. EE synapses), a probability distribution over the parameters of the other synapses (EI, IE, II)? Should we expect more variability in plasticity parameters in biology in dimensions of parameter space that are not constrained by the mean-field equations for stabilizing the network?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their insightful review. We have already addressed the more major concerns above, but elaborate here on the more technical and philosophical points that we couldn’t cover in the common rebuttal:\n\n### 1/ Local optimization vs fSBI: \n\nThe reviewer raised the point that sampling methods such as fSBI may be ill-suited to find precisely-tuned, functional rules. Our response to this concern is multi-pronged. First off, in the unlikely event (see below) that fSBI fails to find at least some finely tuned rules, the rules found here may still provide interesting initializations for local optimization.\n\nHowever, we believe that it is implausible for plasticity rules to be finely tuned, as such fine orchestration would likely lead to catastrophic network-level consequences in the face of perturbation. What’s more, based on current understanding, noisy biochemical processes and sensors could not implement them. We also believe that plasticity rules are unlikely to be unique, given that degeneracy is widespread in biological systems (Prinz et al., Nat. Neur. 2004). These arguments suggest that there should be many functionally interesting rules, instead of a unique solution that a sampling method such as fSBI would indeed struggle to find. Such speculations are supported by the (preliminary) evidence in Fig.R2 that many rules selected for plausibility ascribed a function to a network during a familiarity detection task.\n\nTo provide some context, we turned to SBI because of the difficulty of scaling meta-learning approaches such as those presented by Confavreux et al. NeurIPS 2020 or Jordan et al. eLife 2021 —which consider a few spiking neurons and no recurrence— to large recurrent spiking networks with more flexible plasticity search spaces. In our hands, the added expressive power of such networks and rules made defining a satisfying loss function very difficult. Specifically, we could meta-learn plasticity rules that solved various specific tasks, but the networks would lack many biologically plausible features, instead displaying synchronous and/or regular activity, unrealistically fast weight changes, or unrealistic distributions of firing rates. We could address some of these problems by adding terms to our loss functions, but every metric had blind spots (see main paper Fig.2E,F,G) which meant adding (and having to tune) more and more terms in the loss, which quickly became untenable. Hence we decided to first filter out all the rules that were not producing plausible networks, flexibly, one constraint at a time, implicitly crafting a comprehensive loss function on the fly, without the need for fine-tuning.\n\nWe would also like to emphasize that Confavreux et al. don’t investigate any function beyond stability in their spiking neuron experiments.\n\n### 2/ Meta-learning vs analytical approaches: \n\nThe reviewer raised the concern that meta-learning might not have the potential to discover truly novel rules. We believe that our results answer this question as we found rules that are not predicted by mean field assumptions. Of course, we believe that both approaches must complement one another, and our results may inspire refinements to mean-field theory to allow for the inclusion of discovered rules. Conversely, the choice of the plasticity search space should also be informed by analytical studies proposing promising plasticity mechanisms.\n\n### 3/ Extending the plasticity search space, scaling of fSBI:\n\nThe reviewer suggested extending our study to multi-compartmental models and correspondingly more detailed plasticity models. This is an interesting avenue for future work, but unfortunately out of scope. However, in line with this idea, we highlight that bursts have been implemented previously as high powers of synaptic traces (Zenke et al. Nat. Comm. 2015). Thus, our MLP search space could, in theory, detect and use bursts to craft weight updates. \n\nConsidering the scaling of the method with the number of parameters of the plasticity search space, it is computationally expensive to change search space, hence our focus on two search spaces. Previous applications of SBI were successful up to 30 parameters (Cranmer et al. PNAS 2020, Goncalves et al. eLife 2020), and recent developments have increased this bound in specific settings to a few 1000 parameters (Ramesh et al. ICLR 2022). One of our proposed fixes for this dilemma was to keep some of the MLP parameters fixed, so the effective dimension is low, while the rule search space dimension remains as high as possible. The definition of a flexible yet moderately high-dimensional search space is a key step that will need much more work. This limitation is shared with other meta-learning approaches (Confavreux et al. NeurIPS 2020, Jordan et al. eLife 2021), but we believe we make a valuable contribution towards this goal.\n\n### 4/ What to do with the distributions of rules proposed:\n\nWe agree with the reviewer that unraveling and interpreting the structure we found in the posteriors is a major extension of our work, that is out of scope of this study. However, we are currently working on methods to make sense of the high-dimensional posteriors that come out of fSBI, building on previous studies (Deistler et al., PNAS 2022; Gonçalves et al. eLife 2020). The reviewer’s suggestion to fix one rule (e.g., EE) and look at how it constrains the other rules is a possible analysis (conditional posteriors) and an interesting avenue for future work.\n\n### 5/”Should we expect more variability in plasticity parameters in biology in dimensions of parameter space that are not constrained by the mean-field equations for stabilizing the network?”\n\nThis is an interesting suggestion, and we have begun investigating this, but we currently don’t have clear answers. We will discuss this in the paper, space permitting."}]

Author rebuttal:
We thank the reviewers for their constructive comments and their positive appraisal of our study, calling it "a good study that contributes to the field" (r. vJPD), "a nice extension of simulation-based inference" (r. 5no9), with "insightful” analysis” (r. jNn6), and the discovery of non-mean-field predicted rules deemed "an endorsement of the value of their method” (r. bG1x), and for unanimously recommending acceptance. Below, we discuss the most common questions and comments that have led to additional results and analyses (1 additional main figure and 1 supplementary figure below). We provide further details and additional responses in each individual review.  

## 1/ Choice of metrics and their ranges:

Several of the reviewers commented on subpar reasoning for choices of parameters. Our choices were  informed by –and in line with– a range of modeling and experimental cortical studies [1-13], but we have now realized this was not sufficiently explained. We will add a more detailed discussion to Appendix 5.2. Key points:

Population firing rates $r_{exc}, r_{inh}$ were constrained between 1Hz and 50Hz, in line with a large swatch of literature reporting cortical firing rates of a few Hertz on average [1,8,9]. Notably, we only constrained population averages, individual firing rates could be higher or lower (r. vJPD). 

Maximum average weights $<w_{EE}>, <w_{EI}>, <w_{IE}>, <w_{II}>$ were chosen so that PSPs were smaller than ~3 millivolts, a reasonable max weight according to experimental data [8, 10] (r. vJPD). Similarly, the other weight metrics ($f_{w_{blow}}, w_{creep}$) were chosen to reject biologically unrealistic scenarios, e.g., too many 0 weights or maximum weight $w_{max}$, or weights that changed unrealistically fast.

Cortical activity is broadly asynchronous-irregular [1,2,8,9,4]. Ranges for metrics assessing synchrony/regularity were thus devised with independent Poisson spike trains in mind, with leeway for spatiotemporal correlations, as seen in cortical recordings [11,12,13]. 

We will also make it more apparent in the manuscript that our choices of metrics and ranges are not universal, and a focus on specific brain regions, especially subcortical ones, may necessitate some changes  (r. vJPD). We’d like to highlight that fSBI allows for such changes of metrics and corresponding ranges post hoc, at little computational cost, because we store all spike trains and weight traces (and will make them accessible to the community).

## 2/ Robustness of fSBI rules:

In response to questions regarding the robustness of our results, we ran additional simulations and found that the “plausible” rules from the main paper generalized to different network sizes, sparsity, weight initializations, and ratios of E to I (Fig.R1A, r. 5no9 \& bG1x), although the respective metrics $f_{w_{blow}}$ and $w_{creep}$ were sometimes outside the predefined ranges.

We then tested “plausible” rules on 20min long simulations (versus 2min in the main paper, r. 5no9). Approx. 25% of the polynomial rules still met the conditions on all 15 metrics after 20min (Fig.R1B). Most other rules were  disqualified by our $<w_{IE}>$ and $<w_{II}>$ conditions, which select long-term stable weights, and are thus sensitive to simulation duration. Correspondingly, the II and IE plasticity parameters appeared more refined compared to the starting set (Fig.R1D). Overall, this suggests that the 2min task had properly constrained the EE and EI rules, but was too short to fully constrain the II and IE rules.

Similarly, 35% of the 2min-plausible MLP rules met all conditions in the 20min task. Interestingly, dimensionality reduction of the successful plasticity parameters did not reveal sub-structures, suggesting that the 20min task did not dramatically refine the set of filtered rules (Fig.R1C).

We thank the reviewers for their suggested controls and will add them to the paper. We believe that these experiments further emphasize the advantages of fSBI and its robustness. 

## 3/ Rules with a specific network function:

Reviewer bG1x pointed out that it was unclear whether the candidate rules obtained in this study had the potential to ascribe any interesting network function beyond stabilizing a recurrent spiking network. We thus tested our “plausible” rules on a familiarity detection task (Fig.R2A).

We first tested which of the pre-identified, stable rules met the plausibility conditions during the familiarity detection task (over 80%, further confirming the rules’ robustness, Fig.R2C). Next, we introduced new metrics that were specific to the task, such as measuring the population activity during novel/familiar stimulus presentation. Novel and familiar stimuli were of identical magnitude, thus eliciting the same population response in the absence of plasticity. Interestingly, most polynomial rules responded more strongly to novel stimuli after learning (Fig.R2B), while MLP rules exhibited a more balanced distribution of preferences towards familiar, novel, or neither (Fig.R2D). Overall, many polynomial and MLP rules filtered for plausibility conferred a function —novelty or familiarity detection— to the network they were embedded in. Our new results indicate that rules with interesting network functions beyond mere stabilization may not be so rare in the set of filtered rules presented in this study.

## References: 
[1] Dayan and Abbott, 2001

[2] Brunel, Jour. of Comp. Neur., 2000

[3] Vogels, Rajan and Abbott, Ann. Rev. Neur., 2005

[4] Renart, …, Harris, Science, 2010

[5] Vogels, …, Gerstner, Science, 2011

[6] Zenke, Agnes and Gerstner, Nat. Com., 2015

[7] Sanzeni, Histed and Brunel, Phys. Rev. X, 2023

[8] Shadlen and Newsome, Cur. Op. in Neurob., 1994

[9] Mainen and Sejnowski, Science, 1995

[10] Song, …, Chklovskii, Plos Bio., 2005

[11] Okun and Lampl, Nat. Neur., 2008

[12] Pillow, …, Simoncelli, Nature, 2008

[13] Cohen and Kohn, Nat. Neur., 2011

[14] Prinz, Bucher and Marder, Nat. Neur., 2004



Meta-review: The authors propose a simulation-based method to filter synaptic plasticity rules within biological constraints in large recurrent networks of spiking neurons.

The reviewers agree that the manuscript is well-written and technically sound. In general, the investigation of synaptic plasticity rules is a key problem in (computational) neuroscience. The reviewers agree that this work provides a strong contribution to the current state in this research. The manuscript derives interesting conclusions, also when comparing obtained rules with theoretically grounded mean-field approximations. Conclusions are well-supported by the simulation results.

One potential weakness is that the uncovered rules do not perform interesting functions beyond stabilization of networks. An extension in this direction would be interesting future work.

In summary, an interesting and well-performed study.
Decision: Accept (poster)