arXiv subjects

Srdjan Ostojic

Publications and source records attributed to Srdjan Ostojic.

At least 19 recordsLinked to original sources

arXiv2026

Balancing structure and randomness: maximum entropy networks for context-dependent computations

Understanding how network function constrains neural connectivity is a central challenge in neuroscience. An influential approach is to train neural networks with gradient descent on cognitive tasks and characterize the resulting connectivity. A key limitation is that the resulting structure depends on the details of the training procedure. Here we propose a complementary normative approach based on the maximum entropy principle for network connectivity, independent of any particular learning algorithm. We describe connectivity as a probability distribution over single-neuron weights, express task requirements as constraints on this distribution, and determine the unique distribution maximizing Shannon entropy subject to these constraints. A weight scale parameter controls the balance between randomness and task-induced structure. We apply this framework to context-dependent input-selection tasks in 2-layer feed-forward networks, and show that maximum entropy inference becomes analytically tractable by mapping nonlinear networks onto gain-modulated linear models. Starting from an a priori homogeneous distribution, we find that maximizing entropy under task constraints leads to the emergence of populations of neurons, each defined by its pattern of contextual gain modulation. Increasing the number of contexts drives a transition from context-specialized to unspecialized, random populations. Increasing the weight scale drives a parallel transition from structured to random stimulus selectivity. Strikingly, this maximum entropy connectivity matches both qualitatively and quantitatively the structure of networks trained with gradient descent across different learning regimes. Our results suggest that the interplay between task constraints and entropy maximization provides a fundamental principle for understanding the relationship between structure and function in neural networks.

q-bio.NC↗

arXiv2025

Shaping manifolds in equivariant recurrent neural networks

Recordings of increasingly large neural populations have revealed that the firing of individual neurons is highly coordinated. When viewed in the space of all possible patterns, the collective activity forms non-linear structures called neural manifolds. Because such structures are observed even at rest or during sleep, an important hypothesis is that activity manifolds may correspond to continuous attractors shaped by recurrent connectivity between neurons. Classical models of recurrent networks have shown that continuous attractors can be generated by specific symmetries in the connectivity. Although a variety of attractor network models have been studied, general principles linking network connectivity and the geometry of attractors remain to be formulated. Here, we address this question by using group representation theory to formalize the relationship between the symmetries in recurrent connectivity and the resulting fixed-point manifolds. We start by revisiting the classical ring model, a continuous attractor network generating a circular manifold. Interpreting its connectivity as a circular convolution, we draw a parallel with feed-forward CNNs. Building on principles of geometric deep learning, we then generalize this architecture to a broad range of symmetries using group representation theory. Specifically, we introduce a new class of equivariant RNNs, where the connectivity is based on group convolution. Using the group Fourier transform, we reduce such networks to low-rank models, giving us a low-dimensional description that can be fully analyzed to determine the symmetry, dimensionality and stability of fixed-point manifolds. Our results underline the importance of stability considerations: for a connectivity with a given symmetry, depending on parameters, several manifolds with different symmetry subgroups can coexist, some stable and others consisting of saddle points.

q-bio.NC↗

arXiv2025

Identifying the impact of local connectivity patterns on dynamics in excitatory-inhibitory networks

Networks of excitatory and inhibitory (EI) neurons form a canonical circuit in the brain. Seminal theoretical results on dynamics of such networks are based on the assumption that synaptic strengths depend on the type of neurons they connect, but are otherwise statistically independent. Recent synaptic physiology datasets however highlight the prominence of specific connectivity patterns that go well beyond what is expected from independent connections. While decades of influential research have demonstrated the strong role of the basic EI cell type structure, to which extent additional connectivity features influence dynamics remains to be fully determined. Here we examine the effects of pairwise connectivity motifs on the linear dynamics in EI networks using an analytical framework that approximates the connectivity in terms of low-rank structures. This low-rank approximation is based on a mathematical derivation of the dominant eigenvalues of the connectivity matrix and predicts the impact on responses to external inputs of connectivity motifs and their interactions with cell-type structure. Our results reveal that a particular pattern of connectivity, chain motifs, have a much stronger impact on dominant eigenmodes than other pairwise motifs. An overrepresentation of chain motifs induces a strong positive eigenvalue in inhibition-dominated networks and generates a potential instability that requires revisiting the classical excitation-inhibition balance criteria. Examining effects of external inputs, we show that chain motifs can on their own induce paradoxical responses where an increased input to inhibitory neurons leads to a decrease in their activity due to the recurrent feedback. These findings have direct implications for the interpretation of experiments in which responses to optogenetic perturbations are measured and used to infer the dynamical regime of cortical circuits.

q-bio.NC↗

arXiv2024

Aligned and oblique dynamics in recurrent neural networks

The relation between neural activity and behaviorally relevant variables is at the heart of neuroscience research. When strong, this relation is termed a neural representation. There is increasing evidence, however, for partial dissociations between activity in an area and relevant external variables. While many explanations have been proposed, a theoretical framework for the relationship between external and internal variables is lacking. Here, we utilize recurrent neural networks (RNNs) to explore the question of when and how neural dynamics and the network's output are related from a geometrical point of view. We find that training RNNs can lead to two dynamical regimes: dynamics can either be aligned with the directions that generate output variables, or oblique to them. We show that the choice of readout weight magnitude before training can serve as a control knob between the regimes, similar to recent findings in feedforward networks. These regimes are functionally distinct. Oblique networks are more heterogeneous and suppress noise in their output directions. They are furthermore more robust to perturbations along the output directions. Crucially, the oblique regime is specific to recurrent (but not feedforward) networks, arising from dynamical stability considerations. Finally, we show that tendencies towards the aligned or the oblique regime can be dissociated in neural recordings. Altogether, our results open a new perspective for interpreting neural activity by relating network dynamics and their output.

q-bio.NC↗

arXiv2023

Brief technical note on linearizing recurrent neural networks (RNNs) before vs after the pointwise nonlinearity

Linearization of the dynamics of recurrent neural networks (RNNs) is often used to study their properties. The same RNN dynamics can be written in terms of the ``activations" (the net inputs to each unit, before its pointwise nonlinearity) or in terms of the ``activities" (the output of each unit, after its pointwise nonlinearity); the two corresponding linearizations are different from each other. This brief and informal technical note describes the relationship between the two linearizations, between the left and right eigenvectors of their dynamics matrices, and shows that some context-dependent effects are readily apparent under linearization of activity dynamics but not linearization of activation dynamics.

cs.LG↗

arXiv2023

The computational role of structure in neural activity and connectivity

One major challenge of neuroscience is finding interesting structures in a seemingly disorganized neural activity. Often these structures have computational implications that help to understand the functional role of a particular brain area. Here we outline a unified approach to characterize these structures by inspecting the representational geometry and the modularity properties of the recorded activity, and show that this approach can also reveal structures in connectivity. We start by setting up a general framework for determining geometry and modularity in activity and connectivity and relating these properties with computations performed by the network. We then use this framework to review the types of structure found in recent works on model networks performing three classes of computations.

q-bio.NC↗

arXiv2021

Probing the relationship between linear dynamical systems and low-rank recurrent neural network models

A large body of work has suggested that neural populations exhibit low-dimensional dynamics during behavior. However, there are a variety of different approaches for modeling low-dimensional neural population activity. One approach involves latent linear dynamical system (LDS) models, in which population activity is described by a projection of low-dimensional latent variables with linear dynamics. A second approach involves low-rank recurrent neural networks (RNNs), in which population activity arises directly from a low-dimensional projection of past activity. Although these two modeling approaches have strong similarities, they arise in different contexts and tend to have different domains of application. Here we examine the precise relationship between latent LDS models and linear low-rank RNNs. When can one model class be converted to the other, and vice versa? We show that latent LDS models can only be converted to RNNs in specific limit cases, due to the non-Markovian property of latent LDS models. Conversely, we show that linear RNNs can be mapped onto LDS models, with latent dimensionality at most twice the rank of the RNN.

q-bio.NC↗

arXiv2021

Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity

The ongoing exponential rise in recording capacity calls for new approaches for analysing and interpreting neural data. Effective dimensionality has emerged as an important property of neural activity across populations of neurons, yet different studies rely on different definitions and interpretations of this quantity. Here we focus on intrinsic and embedding dimensionality, and discuss how they might reveal computational principles from data. Reviewing recent works, we propose that the intrinsic dimensionality reflects information about the latent variables encoded in collective activity, while embedding dimensionality reveals the manner in which this information is processed. We conclude by highlighting the role of network models as an ideal substrate for testing more specifically various hypotheses on the computational principles reflected through intrinsic and embedding dimensionality.

q-bio.NC↗

arXiv2021

The interplay between randomness and structure during learning in RNNs

Recurrent neural networks (RNNs) trained on low-dimensional tasks have been widely used to model functional biological networks. However, the solutions found by learning and the effect of initial connectivity are not well understood. Here, we examine RNNs trained using gradient descent on different tasks inspired by the neuroscience literature. We find that the changes in recurrent connectivity can be described by low-rank matrices, despite the unconstrained nature of the learning algorithm. To identify the origin of the low-rank structure, we turn to an analytically tractable setting: training a linear RNN on a simplified task. We show how the low-dimensional task structure leads to low-rank changes to connectivity. This low-rank structure allows us to explain and quantify the phenomenon of accelerated learning in the presence of random initial connectivity. Altogether, our study opens a new perspective to understanding trained RNNs in terms of both the learning process and the resulting network structure.

q-bio.NC↗

arXiv2020

Shaping dynamics with multiple populations in low-rank recurrent networks

An emerging paradigm proposes that neural computations can be understood at the level of dynamical systems that govern low-dimensional trajectories of collective neural activity. How the connectivity structure of a network determines the emergent dynamical system however remains to be clarified. Here we consider a novel class of models, Gaussian-mixture low-rank recurrent networks, in which the rank of the connectivity matrix and the number of statistically-defined populations are independent hyper-parameters. We show that the resulting collective dynamics form a dynamical system, where the rank sets the dimensionality and the population structure shapes the dynamics. In particular, the collective dynamics can be described in terms of a simplified effective circuit of interacting latent variables. While having a single, global population strongly restricts the possible dynamics, we demonstrate that if the number of populations is large enough, a rank-R network can approximate any R-dimensional dynamical system.

q-bio.NC↗

arXiv2019

Dynamics of random recurrent networks with correlated low-rank structure

A given neural network in the brain is involved in many different tasks. This implies that, when considering a specific task, the network's connectivity contains a component which is related to the task and another component which can be considered random. Understanding the interplay between the structured and random components, and their effect on network dynamics and functionality is an important open question. Recent studies addressed the co-existence of random and structured connectivity, but considered the two parts to be uncorrelated. This constraint limits the dynamics and leaves the random connectivity non-functional. Algorithms that train networks to perform specific tasks typically generate correlations between structure and random connectivity. Here we study nonlinear networks with correlated structured and random components, assuming the structure to have a low rank. We develop an analytic framework to establish the precise effect of the correlations on the eigenvalue spectrum of the joint connectivity. We find that the spectrum consists of a bulk and multiple outliers, whose location is predicted by our theory. Using mean-field theory, we show that these outliers directly determine both the fixed points of the system and their stability. Taken together, our analysis elucidates how correlations allow structured and random connectivity to synergistically extend the range of computations available to networks.

q-bio.NC↗

arXiv2019

Coding with transient trajectories in recurrent neural networks

Following a stimulus, the neural response typically strongly varies in time and across neurons before settling to a steady-state. While classical population coding theory disregards the temporal dimension, recent works have argued that trajectories of transient activity can be particularly informative about stimulus identity and may form the basis of computations through dynamics. Yet the dynamical mechanisms needed to generate a population code based on transient trajectories have not been fully elucidated. Here we examine transient coding in a broad class of high-dimensional linear networks of recurrently connected units. We start by reviewing a well-known result that leads to a distinction between two classes of networks: networks in which all inputs lead to weak, decaying transients, and networks in which specific inputs elicit strongly amplified transient responses and are mapped onto orthogonal output states during the dynamics. Theses two classes are simply distinguished based on the spectrum of the symmetric part of the connectivity matrix. For the second class of networks, which is a sub-class of non-normal networks, we provide a procedure to identify transiently amplified inputs and the corresponding readouts. We first apply these results to standard randomly-connected and two-population networks. We then build minimal, low-rank networks that robustly implement trajectories mapping a specific input onto a specific output state. Finally, we demonstrate that the capacity of the obtained networks increases proportionally with their size.

q-bio.NC↗

arXiv2019

Contrasting the effects of adaptation and synaptic filtering on the timescales of dynamics in recurrent networks

Neural activity exhibits a vast range of timescales that can be several fold larger than the membrane time constant of individual neurons. Two types of mechanisms have been proposed to explain this conundrum. One possibility is that large timescales are generated by a network mechanism based on positive feedback, but this hypothesis requires fine-tuning of the synaptic connections. A second possibility is that large timescales in the neural dynamics are inherited from large timescales of underlying biophysical processes, two prominent candidates being adaptive ionic currents and synaptic transmission. How the timescales of these processes influence the timescale of the network dynamics has however not been fully explored. To address this question, we analyze large networks of randomly connected excitatory and inhibitory units with additional degrees of freedom that correspond to adaptation or synaptic filtering. We determine the fixed points of the systems, their stability to perturbations and the corresponding dynamical timescales. Furthermore, we apply dynamical mean field theory to study the temporal statistics of the activity in the fluctuating regime, and examine how the adaptation and synaptic timescales transfer from individual units to the whole population. Our overarching finding is that synaptic filtering and adaptation in single neurons have very different effects at the network level. Unexpectedly, the macroscopic network dynamics do not inherit the large timescale present in adaptive currents. In contrast, the timescales of network activity increase proportionally to the time constant of the synaptic filter. Altogether, our study demonstrates that the timescales of different biophysical processes have different effects on the network level, so that the slow processes within individual neurons do not necessarily induce slow activity in large recurrent neural networks.

q-bio.NC↗

arXiv2019

A geometrical analysis of global stability in trained feedback networks

Recurrent neural networks have been extensively studied in the context of neuroscience and machine learning due to their ability to implement complex computations. While substantial progress in designing effective learning algorithms has been achieved in the last years, a full understanding of trained recurrent networks is still lacking. Specifically, the mechanisms that allow computations to emerge from the underlying recurrent dynamics are largely unknown. Here we focus on a simple, yet underexplored computational setup: a feedback architecture trained to associate a stationary output to a stationary input. As a starting point, we derive an approximate analytical description of global dynamics in trained networks which assumes uncorrelated connectivity weights in the feedback and in the random bulk. The resulting mean-field theory suggests that the task admits several classes of solutions, which imply different stability properties. Different classes are characterized in terms of the geometrical arrangement of the readout with respect to the input vectors, defined in the high-dimensional space spanned by the network population. We find that such approximate theoretical approach can be used to understand how standard training techniques implement the input-output task in finite-size feedback networks. In particular, our simplified description captures the local and the global stability properties of the target solution, and thus predicts training performance.

q-bio.NC↗

arXiv2018

Linking connectivity, dynamics and computations in low-rank recurrent neural networks

Large scale neural recordings have established that the transformation of sensory stimuli into motor outputs relies on low-dimensional dynamics at the population level, while individual neurons exhibit complex selectivity. Understanding how low-dimensional computations on mixed, distributed representations emerge from the structure of the recurrent connectivity and inputs to cortical networks is a major challenge. Here, we study a class of recurrent network models in which the connectivity is a sum of a random part and a minimal, low-dimensional structure. We show that, in such networks, the dynamics are low dimensional and can be directly inferred from connectivity using a geometrical approach. We exploit this understanding to determine minimal connectivity required to implement specific computations, and find that the dynamical range and computational capacity quickly increase with the dimensionality of the connectivity structure. This framework produces testable experimental predictions for the relationship between connectivity, low-dimensional dynamics and computational features of recorded neurons.

q-bio.NC↗

arXiv2018

Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks

Networks of randomly connected neurons are among the most popular models in theoretical neuroscience. The connectivity between neurons in the cortex is however not fully random, the simplest and most prominent deviation from randomness found in experimental data being the overrepresentation of bidirectional connections among pyramidal cells. Using numerical and analytical methods, we investigated the effects of partially symmetric connectivity on dynamics in networks of rate units. We considered the two dynamical regimes exhibited by random neural networks: the weak-coupling regime, where the firing activity decays to a single fixed point unless the network is stimulated, and the strong-coupling or chaotic regime, characterized by internally generated fluctuating firing rates. In the weak-coupling regime, we computed analytically for an arbitrary degree of symmetry the auto-correlation of network activity in presence of external noise. In the chaotic regime, we performed simulations to determine the timescale of the intrinsic fluctuations. In both cases, symmetry increases the characteristic asymptotic decay time of the autocorrelation function and therefore slows down the dynamics in the network.

q-bio.NC↗

arXiv2017

Intrinsically-generated fluctuating activity in excitatory-inhibitory networks

Recurrent networks of non-linear units display a variety of dynamical regimes depending on the structure of their synaptic connectivity. A particularly remarkable phenomenon is the appearance of strongly fluctuating, chaotic activity in networks of deterministic, but randomly connected rate units. How this type of intrinsi- cally generated fluctuations appears in more realistic networks of spiking neurons has been a long standing question. To ease the comparison between rate and spiking networks, recent works investigated the dynami- cal regimes of randomly-connected rate networks with segregated excitatory and inhibitory populations, and firing rates constrained to be positive. These works derived general dynamical mean field (DMF) equations describing the fluctuating dynamics, but solved these equations only in the case of purely inhibitory networks. Using a simplified excitatory-inhibitory architecture in which DMF equations are more easily tractable, here we show that the presence of excitation qualitatively modifies the fluctuating activity compared to purely inhibitory networks. In presence of excitation, intrinsically generated fluctuations induce a strong increase in mean firing rates, a phenomenon that is much weaker in purely inhibitory networks. Excitation moreover induces two different fluctuating regimes: for moderate overall coupling, recurrent inhibition is sufficient to stabilize fluctuations, for strong coupling, firing rates are stabilized solely by the upper bound imposed on activity, even if inhibition is stronger than excitation. These results extend to more general network architectures, and to rate networks receiving noisy inputs mimicking spiking activity. Finally, we show that signatures of the second dynamical regime appear in networks of integrate-and-fire neurons.

q-bio.NC↗

arXiv2016

Instability to a heterogeneous oscillatory state in randomly connected recurrent networks with delayed interactions

Oscillatory dynamics are ubiquitous in biological networks. Possible sources of oscillations are well understood in low-dimensional systems, but have not been fully explored in high-dimensional networks. Here we study large networks consisting of randomly coupled rate units. We identify a novel type of bifurcation in which a continuous part of the eigenvalue spectrum of the linear stability matrix crosses the instability line at non-zero-frequency. This bifurcation occurs when the interactions are delayed and partially anti-symmetric, and leads to a heterogeneous oscillatory state in which oscillations are apparent in the activity of individual units, but not on the population-average level.

cond-mat.dis-nn↗