Dimensionality reduction methods are widely used in neuroscience to investigate two complementary aspects of neural activity: the distribution of single-neuron functional properties and the low-dimensional collective dynamics of population activity. However, how do these two aspects of neural activity relate to the structure of the underlying neural circuit? In this work, we connect circuit structure, single-neuron functional properties, and emerging low-dimensional dynamics in spiking recurrent network models. Our models explain how topologically distinct circuit structures can produce equivalent low-dimensional dynamics. Despite this degeneracy, we find that circuit structure imposes specific constraints on both the low-dimensional dynamics of population activity and the distribution of single-neuron functional properties. These constraints yield simple criteria for comparing network models with observed neural activity. Our modeling framework not only links classical models of cortical circuits to the more recent notion of neural manifolds but also paves the way for designing tractable models of population dynamics that are better aligned with neural recordings.

This work studies the mean-field limit of large networks of interacting stochastic leaky integrate-and-fire (LIF) neurons subject to short-term synaptic depression (STD). The macroscopic dynamics of this system is governed by a two-dimensional, non-linear McKean-Vlasov equation that couples the evolution of the neurons' membrane potentials with a synaptic depression variable. We investigate the long-time behavior of this limit system. To this end, we introduce an auxiliary linearized Markov process by freezing the interaction non-linearity to a constant. By exploiting the regeneration of the membrane potential at spike times, we are able to explicitly compute the conditional expectation of the synaptic depression variable, conditionally on the potential value, under the invariant measure of this two-dimensional linear process. This is a crucial ingredient to study time-dependent local perturbations thereof. As a consequence we are able to identify an analytic criterion guaranteeing the local stability of any invariant probability measure of the fully non-linear system. This stability criterion is formulated in terms of the zeros of the Laplace transform of a specific linear response function. Finally, we provide numerical examples demonstrating that the two-dimensional framework induces a richer spectrum of long-time dynamics than purely one-dimensional models. For example, synaptic depression can lead to low-frequency oscillations around a unique, unstable invariant measure where the oscillations are much slower than the neurons' firing rates.

Neurons primarily communicate through the emission of action potentials, or spikes. To generate a spike, a neuron's membrane potential must cross a defined threshold. Does this spiking mechanism inherently prevent neurons from transmitting their subthreshold membrane potential fluctuations to other neurons? We prove that, in theory, it does not. The subthreshold membrane potential fluctuations of a presynaptic population of spiking neurons can be perfectly transmitted to a downstream population of neurons. Mathematically, this surprising result is an example of concentration phenomenon in high dimensions.

Computation in recurrent networks of neurons has been hypothesized to occur at the level of low-dimensional latent dynamics, both in artificial systems and in the brain. This hypothesis seems at odds with evidence from large-scale neuronal recordings in mice showing that neuronal population activity is high-dimensional. To demonstrate that low-dimensional latent dynamics and high-dimensional activity can be two sides of the same coin, we present an analytically solvable recurrent neural network (RNN) model whose dynamics can be exactly reduced to a low-dimensional dynamical system, but generates an activity manifold that has a high linear embedding dimension. This raises the question: Do low-dimensional latents explain the high-dimensional activity observed in mouse visual cortex? Spectral theory tells us that the covariance eigenspectrum alone does not allow us to recover the dimensionality of the latents, which can be low or high, when neurons are nonlinear. To address this indeterminacy, we develop Neural Cross-Encoder (NCE), an interpretable, nonlinear latent variable modeling method for neuronal recordings, and find that high-dimensional neuronal responses to drifting gratings and spontaneous activity in visual cortex can be reduced to low-dimensional latents, while the responses to natural images cannot. We conclude that the high-dimensional activity measured in certain conditions, such as in the absence of a stimulus, is explained by low-dimensional latents that are nonlinearly processed by individual neurons.

The activity of visual cortical neurons forms a population code representing image stimuli. There is, however, a discrepancy between our understanding of this code at the single-cell and population levels: direct measurements indicate the population code is high-dimensional, but established models of single-cell tuning give rise to low-dimensional codes. We reconciled this discrepancy by developing an AI science system to find a new parsimonious, interpretable equation for visual cortical orientation tuning. Candidate equations were expressed as short computer programs and evolved by Large Language Models (LLMs) using graphical diagnostics. The resulting equation not only improved single-cell fits, but also accurately modelled the population code’s high-dimensional geometry. A novel parameter of the AI-discovered equation, which controls single-cell tuning smoothness, gives rise to high-dimensional population codes. The same parameter drives high-dimensional coding in head-direction cells, suggesting a common coding strategy across brain regions. We used this equation to hypothesize a circuit mechanism generating high-dimensional population codes, and to demonstrate the advantages of these codes in a simulated hyperacuity task. These results show that tuning smoothness has a key role in controlling population code geometry, and demonstrate how AI equation discovery can deliver interpretable models accelerating scientific understanding in neuroscience and beyond.

The visual cortex contains millions of neurons, whose combined activity forms a population code representing visual stimuli. There is, however, a discrepancy between our understanding of this code at the single neuron and population levels: direct measurements suggest that the population code is high-dimensional, but current understanding of single-cell tuning curves implies low dimensionality. To reconcile this discrepancy, we used AI to find a new parsimonious, interpretable equation for tuning curves to oriented stimuli. Candidate equations were expressed as short computer programs, and evolved using Large Language Models (LLMs) to improve their fits to single-cell tuning. This resulted in an equation that not only improved single-cell fits, but also accurately modelled the population code’s high-dimensional structure, even though population dimensionality was not the AI system’s objective. This high-dimensionality occurred because the AI-derived tuning curves were not smooth at their peaks, which we proved is mathematically required for high-dimensional population coding of low-dimensional stimuli. We used the AI-generated tuning equation to demonstrate the advantages of high-dimensional codes in a simulated hyperacuity task, and showed that non-smooth peaks are also present in other brain systems, leading to high-dimensional population coding in head direction cells. These results suggest that the smoothness of neuronal tuning curves has an important role in setting the dimensionality of the population code, and demonstrate how AI equation discovery can accelerate scientific theory building in neuroscience and beyond. ### Competing Interest Statement The authors have declared no competing interest. UKRI, EP/X022366/1, BB/W019884/1 Wellcome Trust, 223144/Z/21/Z, 219627/Z/19/Z ERC, 101097874 Gatsby Charitable Foundation, GAT3755 Royal Society, NIF\R1\231927 Swiss National Science Foundation, 222150

The dynamics of spatially-structured networks of N interacting stochastic neurons can be described by deterministic population equations in the mean-field limit. While this is known, a general question has remained unanswered: does synaptic weight scaling suffices, by itself, to guarantee the convergence of network dynamics to a deterministic population equation, even when networks are not assumed to be homogeneous or spatially structured? In this work, we consider networks of stochastic integrate-and-fire neurons with arbitrary synaptic weights satisfying a O(1/N) scaling condition. Borrowing results from the theory of dense graph limits, or graphons, we prove that, as N→∞, and up to the extraction of a subsequence, the empirical measure of the neurons' membrane potentials converges to the solution of a spatially-extended mean-field partial differential equation (PDE). Our proof requires analytical techniques that go beyond standard propagation of chaos methods. In particular, we introduce a weak metric that depends on the dense graph limit kernel and we show how the weak convergence of the initial data can be obtained by propagating the regularity of the limit kernel along the dual-backward equation associated with the spatially-extended mean-field PDE. Overall, this result invites us to reinterpret spatially-extended population equations as universal mean-field limits of networks of neurons with O(1/N) synaptic weight scaling.

Can spiking neural networks (SNNs) approximate the dynamics of recurrent neural networks? Arguments in classical mean-field theory based on laws of large numbers provide a positive answer when each neuron in the network has many "duplicates", i.e., other neurons with almost perfectly correlated inputs. Using a disordered network model that guarantees the absence of duplicates, we show that duplicate-free SNNs can converge to recurrent neural networks, thanks to the concentration of measure phenomenon. This result reveals a general mechanism underlying the emergence of rate-based dynamics in large SNNs.

While large-scale recording techniques indicate that the activity of heterogeneous neuronal populations lies on low-dimensional “neural manifolds”, it has remained challenging to reconcile this picture with the classical view of precisely tuned neurons interacting with each other in some ordered circuit structure. Using a modelling approach, we provide a conceptual, yet mathematically precise, link between these two contrasting views. We first show that there is no unique relationship between the circuit structure and the emergent low-dimensional dynamics that characterise the population activity. We then propose a method for retrieving the circuit structure from recordings of the population activity and test it on artificial data. Our approach provides not only a unifying framework for circuit and field models on one side, and low-rank networks on the other side, but also opens the perspective to identify principles of circuit structure from large-scale recordings.### Competing Interest StatementThe authors have declared no competing interest.

Population equations for infinitely large networks of spiking neurons have a long tradition in theoretical neuroscience. In this work, we analyze a recent generalization of these equations to populations of finite size, which takes the form of a nonlinear stochastic integral equation. We prove that, in the case of leaky integrate-and-fire (LIF) neurons with escape noise and for a slightly simplified version of the model, the equation is well-posed and stable in the sense of Br\'emaud-Massouli\'e. The proof combines methods from Markov processes taking values in the space of positive measures and nonlinear Hawkes processes. For applications, we also provide efficient simulation algorithms.

We propose a mean-field model of interacting point processes where each process has a memory of the time elapsed since its last event (age) and its recent past (leaky memory), generalizing Age-dependent Hawkes processes. The model is motivated by interacting nonlinear Hawkes processes with Markovian self-interaction and networks of spiking neurons with adaptation and short-term synaptic plasticity. By proving propagation of chaos and using a path integral representation for the law of the limit process, we show that, in the mean-field limit, the empirical measure of the system follows a multidimensional nonlocal transport equation.

Bottom-up models of functionally relevant patterns of neural activity provide an explicit link between neuronal dynamics and computation. A prime example of functional activity patterns are propagating bursts of place-cell activities called hippocampal replay, which is critical for memory consolidation. The sudden and repeated occurrences of these burst states during ongoing neural activity suggest metastable neural circuit dynamics. As metastability has been attributed to noise and/or slow fatigue mechanisms, we propose a concise mesoscopic model which accounts for both. Crucially, our model is bottom-up: it is analytically derived from the dynamics of finite-size networks of Linear-Nonlinear Poisson neurons with short-term synaptic depression. As such, noise is explicitly linked to stochastic spiking and network size, and fatigue is explicitly linked to synaptic dynamics. To derive the mesoscopic model, we first consider a homogeneous spiking neural network and follow the temporal coarse-graining approach of Gillespie to obtain a “chemical Langevin equation”, which can be naturally interpreted as a stochastic neural mass model. The Langevin equation is computationally inexpensive to simulate and enables a thorough study of metastable dynamics in classical setups (population spikes and Up-Down-states dynamics) by means of phase-plane analysis. An extension of the Langevin equation for small network sizes is also presented. The stochastic neural mass model constitutes the basic component of our mesoscopic model for replay. We show that the mesoscopic model faithfully captures the statistical structure of individual replayed trajectories in microscopic simulations and in previously reported experimental data. Moreover, compared to the deterministic Romani-Tsodyks model of place-cell dynamics, it exhibits a higher level of variability regarding order, direction and timing of replayed trajectories, which seems biologically more plausible and could be functionally desirable. This variability is the product of a new dynamical regime where metastability emerges from a complex interplay between finite-size fluctuations and local fatigue.

Can we use spiking neural networks (SNN) as generative models of multi-neuronal recordings, while taking into account that most neurons are unobserved? Modeling the unobserved neurons with large pools of hidden spiking neurons leads to severely underconstrained problems that are hard to tackle with maximum likelihood estimation. In this work, we use coarse-graining and mean-field approximations to derive a bottom-up, neuronally-grounded latent variable model (neuLVM), where the activity of the unobserved neurons is reduced to a low-dimensional mesoscopic description. In contrast to previous latent variable models, neuLVM can be explicitly mapped to a recurrent, multi-population SNN, giving it a transparent biological interpretation. We show, on synthetic spike trains, that a few observed neurons are sufficient for neuLVM to perform efficient model inversion of large SNNs, in the sense that it can recover connectivity parameters, infer single-trial latent population activity, reproduce ongoing metastable dynamics, and generalize when subjected to perturbations mimicking photo-stimulation.

We study the asymptotic stability of a two-dimensional mean-field equation, which takes the form of a nonlocal transport equation and generalizes the time-elapsed neuron network model by the inclusion of a leaky memory variable. This additional variable can represent a slow fatigue mechanism, like spike frequency adaptation or short-term synaptic depression. Even though two-dimensional models are known to have emergent behaviors, like population bursts, which are not observed in standard one-dimensional models, we show that in the weak connectivity regime, two-dimensional models behave like one-dimensional models, i.e. they relax to a unique stationary state. The proof is based on an application of Harris' ergodic theorem and a perturbation argument, adapted to the case of a multidimensional equation with delays.

Coarse-graining microscopic models of biological neural networks to obtain mesoscopic models of neural activities is an essential step towards multi-scale models of the brain. Here, we extend a recent theory for mesoscopic population dynamics with static synapses to the case of dynamic synapses exhibiting short-term plasticity (STP). The extended theory offers an approximate mean-field dynamics for the synaptic input currents arising from populations of spiking neurons and synapses undergoing Tsodyks-Markram STP. The approximate mean-field dynamics accounts for both finite number of synapses and correlation between the two synaptic variables of the model (utilization and available resources) and its numerical implementation is simple. Comparisons with Monte Carlo simulations of the microscopic model show that in both feedforward and recurrent networks, the mesoscopic mean-field model accurately reproduces the first- and second-order statistics of the total synaptic input into a postsynaptic neuron and accounts for stochastic switches between Up and Down states and for population spikes. The extended mesoscopic population theory of spiking neural networks with STP may be useful for a systematic reduction of detailed biophysical models of cortical microcircuits to numerically efficient and mathematically tractable mean-field models.

The adult dentate gyrus produces new neurons that morphologically and functionally integrate into the hippocampal network. In the adult brain, most excitatory synapses are ensheathed by astrocytic perisynaptic processes that regulate synaptic structure and function. However, these processes are formed during embryonic or early postnatal development and it is unknown whether astrocytes can also ensheathe synapses of neurons born during adulthood and, if so, whether they play a role in their synaptic transmission. Here, we used a combination of serial-section immuno-electron microscopy, confocal microscopy, and electrophysiology to examine the formation of perisynaptic processes on adult-born neurons. We found that the afferent and efferent synapses of newborn neurons are ensheathed by astrocytic processes, irrespective of the age of the neurons or the size of their synapses. The quantification of gliogenesis and the distribution of astrocytic processes on synapses formed by adult-born neurons suggest that the majority of these processes are recruited from pre-existing astrocytes. Furthermore, the inhibition of astrocytic glutamate re-uptake significantly reduced postsynaptic currents and increased paired-pulse facilitation in adult-born neurons, suggesting that perisynaptic processes modulate synaptic transmission on these cells. Finally, some processes were found intercalated between newly formed dendritic spines and potential presynaptic partners, suggesting that they may also play a structural role in the connectivity of new spines. Together, these results indicate that pre-existing astrocytes remodel their processes to ensheathe synapses of adult-born neurons and participate to the functional and structural integration of these cells into the hippocampal network.

Self-consciousness is based on multisensory signals from the body. In full-body illusion (FBI) experiments, multisensory conflict was used to induce changes in three key aspects of bodily self-consciousness (BSC): self-identification (which body ‘I’ identify with), self-location (where ‘I’ am located), and first-person perspective (from where ‘I’ experience the world; 1PP). Here, we adapted a previous FBI protocol in which visuotactile stroking was administered by a robotic device (tactile stroking) and simultaneously rendered on the back of a virtual body (visual stroking) that participants viewed on a head-mounted display as if filmed from a posterior viewpoint of a camera. We compared the effects of two different visuospatial viewpoints on the FBI and thereby on these key aspects of BSC. During control manipulations, participants saw a no-body object instead of a virtual body (first experiment) or received asynchronous versus synchronous visuotactile stroking (second experiment). Results showed that within-subjects visuospatial viewpoint manipulations affected the subjective 1PP ratings if a virtual body was seen but had no effect for viewing a non-body object. However, visuospatial viewpoint had no effect on self-identification, but depended on the viewed object and visuotactile synchrony. Self-location depended on visuospatial viewpoint (first experiment) and visuotactile synchrony (second experiment). Our results show that the visuospatial viewpoint from which the virtual body is seen during FBIs modulates the subjective 1PP and that such viewpoint manipulations contribute to spatial aspects of BSC. We compare the present data with recent data revealing vestibular contributions to the subjective 1PP and discuss the multisensory nature of BSC and the subjective 1PP.

In three experiments we investigated the effects of visuo-tactile and visuo-vestibular conflict about the direction of gravity on three aspects of bodily self-consciousness: self-identification, self-location, and the experienced direction of the first-person perspective. Robotic visuo-tactile stimulation was administered to 78 participants in three experiments. Additionally, we presented participants with a virtual body as seen from an elevated and downward-directed perspective while they were lying supine and were therefore receiving vestibular and postural cues about an upward-directed perspective. Under these conditions, we studied the effects of different degrees of visuo-vestibular conflict, repeated measurements during illusion induction, and the relationship to a classical measure of visuo-vestibular integration. Extending earlier findings on experimentally induced changes in bodily self-consciousness, we show that self-identification does not depend on the experienced direction of the first-person perspective, whereas self-location does. Changes in bodily self-consciousness depend on visual gravitational signals. Individual differences in the experienced direction of first-person perspective correlated with individual differences in visuo-vestibular integration. Our data reveal important contributions of visuo-vestibular gravitational cues to bodily self-consciousness. In particular we show that the experienced direction of the first-person perspective depends on the integration of visual, vestibular, and tactile signals, as well as on individual differences in idiosyncratic visuo-vestibular strategies.