Dynamical Systems seminars

Enaction plays a central role in the broader fabric of so-called 4E (embodied, embedded, extended, enactive) cognition. Although the origin of the enactive approach is widely dated to the 1991 publication of the book "The Embodied Mind" by Varela, Thompson and Rosch, many of the central ideas trace to much earlier work. Over 40 years ago, the Chilean biologists Humberto Maturana and Francisco Varela put forward the notion of autopoiesis as a way to understand living systems and the phenomena that they generate, including cognition. Varela and others subsequently extended this framework to an enactive approach that places biological autonomy at the foundation of situated and embodied behavior and cognition. I will describe an attempt to place Maturana and Varela's original ideas on a firmer foundation by studying them within the context of a toy model universe, John Conway's Game of Life (GoL) cellular automata. This work has both pedagogical and theoretical goals. Simple concrete models provide an excellent vehicle for introducing some of the core concepts of autopoiesis and enaction and explaining how these concepts fit together into a broader whole. In addition, a careful analysis of such toy models can hone our intuitions about these concepts, probe their strengths and weaknesses, and move the entire enterprise in the direction of a more mathematically rigorous theory. In particular, I will identify the primitive processes that can occur in GoL, show how these can be linked together into mutually-supporting networks that underlie persistent bounded entities, map the responses of such entities to environmental perturbations, and investigate the paths of mutual perturbation that these entities and their environments can undergo.

In most brains across the animal kingdom, brain dynamics can enter pathological states that are recognisable as epileptic seizures. Yet usually, brain operate within certain constraints given through neuronal function and synaptic coupling, that will prevent epileptic seizure dynamics from emerging. In this talk, I will bring together different approaches to identifying how networks in the broadest sense shape brain dynamics. Using illustrative examples from intracranial EEG recordings, disorders characterised by molecular disruption of a single neurotransmitter receptor type, to single-cell recordings of whole-brain activity in the larval zebrafish, I will address three key questions - (1) how does the regionally specific composition of synaptic receptors shape ongoing physiological brain activity; (2) how can disruption of this regionally specific balance result in abnormal brain dynamics; and (3) which cellular patterns underly the transition into an epileptic seizure.

The medial entorhinal cortex (MEC) hosts many of the brain’s circuit elements for spatial navigation and episodic memory, operations that require neural activity to be organized across long durations of experience. While location is known to be encoded by a plethora of spatially tuned cell types in this brain region, little is known about how the activity of entorhinal cells is tied together over time. Among the brain’s most powerful mechanisms for neural coordination are network oscillations, which dynamically synchronize neural activity across circuit elements. In MEC, theta and gamma oscillations provide temporal structure to the neural population activity at subsecond time scales. It remains an open question, however, whether similarly coordination occurs in MEC at behavioural time scales, in the second-to-minute regime. In this talk I will show that MEC activity can be organized into a minute-scale oscillation that entrains nearly the entire cell population, with periods ranging from 10 to 100 seconds. Throughout this ultraslow oscillation, neural activity progresses in periodic and stereotyped sequences. The oscillation sometimes advances uninterruptedly for tens of minutes, transcending epochs of locomotion and immobility. Similar oscillatory sequences were not observed in neighboring parasubiculum or in visual cortex. The ultraslow periodic sequences in MEC may have the potential to couple its neurons and circuits across extended time scales and to serve as a scaffold for processes that unfold at behavioural time scales.

A signature feature of cortical circuits is the irregularity of neuronal firing, which manifests itself in the high temporal variability of spiking and the broad distribution of rates. Theoretical works have shown that this feature emerges dynamically in network models if coupling between cells is strong, i.e. if the mean number of synapses per neuron K is large and synaptic efficacy is of order 1/\sqrt{K}. However, the degree to which these models capture the mechanisms underlying neuronal firing in cortical circuits is not fully understood. Results have been derived using neuron models with current-based synapses, i.e. neglecting the dependence of synaptic current on the membrane potential, and an understanding of how irregular firing emerges in models with conductance-based synapses is still lacking. Moreover, at odds with the nonlinear responses to multiple stimuli observed in cortex, network models with strongly coupled cells respond linearly to inputs. In this talk, I will discuss the emergence of irregular firing and nonlinear response in networks of leaky integrate-and-fire neurons. First, I will show that, when synapses are conductance-based, irregular firing emerges if synaptic efficacy is of order 1/\log(K) and, unlike in current-based models, persists even under the large heterogeneity of connections which has been reported experimentally. I will then describe an analysis of neural responses as a function of coupling strength and show that, while a linear input-output relation is ubiquitous at strong coupling, nonlinear responses are prominent at moderate coupling. I will conclude by discussing experimental evidence of moderate coupling and loose balance in the mouse cortex.

Predictive modeling and dimensionality reduction of functional neuroimaging data have provided rich information about the representations and functional architectures of the human brain. While these approaches have been effective in many cases, we will discuss how neglecting the internal dynamics of the brain (e.g., spontaneous activity, global dynamics, effective connectivity) and its underlying computational principles may hinder our progress in understanding and modeling brain functions. By reexamining evidence from our previous and ongoing work, we will propose new hypotheses and directions for research that consider both internal dynamics and the computational principles that may govern brain processes.

Brains can gracefully weed out irrelevant stimuli to guide behavior. This feat is believed to rely on a progressive selection of task-relevant stimuli across the cortical hierarchy, but the specific across-area interactions enabling stimulus selection are still unclear. Here, we propose that population gating, occurring within A1 but controlled by top-down inputs from mPFC, can support across-area stimulus selection. Examining single-unit activity recorded while rats performed an auditory context-dependent task, we found that A1 encoded relevant and irrelevant stimuli along a common dimension of its neural space. Yet, the relevant stimulus encoding was enhanced along an extra dimension. In turn, mPFC encoded only the stimulus relevant to the ongoing context. To identify candidate mechanisms for stimulus selection within A1, we reverse-engineered low-rank RNNs trained on a similar task. Our analyses predicted that two context-modulated neural populations gated their preferred stimulus in opposite contexts, which we confirmed in further analyses of A1. Finally, we show in a two-region RNN how population gating within A1 could be controlled by top-down inputs from PFC, enabling flexible across-area communication despite fixed inter-areal connectivity.

In this talk, we propose to decipher the activity of neural networks via a “multiply and conquer” approach. This approach considers limit networks made of infinitely many replicas with the same basic neural structure. The key point is that these so-called replica-mean-field networks are in fact simplified, tractable versions of neural networks that retain important features of the finite network structure of interest. The finite size of neuronal populations and synaptic interactions is a core determinant of neural dynamics, being responsible for non-zero correlation in the spiking activity and for finite transition rates between metastable neural states. Theoretically, we develop our replica framework by expanding on ideas from the theory of communication networks rather than from statistical physics to establish Poissonian mean-field limits for spiking networks. Computationally, we leverage our original replica approach to characterize the stationary spiking activity of various network models via reduction to tractable functional equations. We conclude by discussing perspectives about how to use our replica framework to probe nontrivial regimes of spiking correlations and transition rates between metastable neural states.

We recorded and analyzed the population activity of hundreds of neurons in the medial premotor areas (MPC) of rhesus monkeys performing an isochronous tapping task guided by brief flashing stimuli or auditory tones. The animals showed a strong bias towards visual metronomes, with rhythmic tapping that was more precise and accurate than for auditory metronomes. The population dynamics in state space as well as the corresponding neural sequences shared the following properties across modalities: the circular dynamics of the neural trajectories and the neural sequences formed a regenerating loop for every produced interval, producing a relative time representation; the trajectories converged in similar state space at tapping times while the moving bumps restart at this point, resetting the beat-based clock; the tempo of the synchronized tapping was encoded by a combination of amplitude modulation and temporal scaling in the neural trajectories. In addition, the modality induced a displacement in the neural trajectories in auditory and visual subspaces without greatly altering time keeping mechanism. These results suggest that the interaction between the amodal internal representation of pulse within MPC and a modality specific external input generates a neural rhythmic clock whose dynamics define the temporal execution of tapping using auditory and visual metronomes.

A major challenge for causal inference from time-series data is the trade-off between computational feasibility and accuracy. Motivated by process motifs for lagged covariance in an autoregressive model with slow mean-reversion, we propose to infer networks of causal relations via pairwise edge measure (PEMs) that one can easily compute from lagged correlation matrices. Motivated by contributions of process motifs to covariance and lagged variance, we formulate two PEMs that correct for confounding factors and for reverse causation. To demonstrate the performance of our PEMs, we consider network interference from simulations of linear stochastic processes, and we show that our proposed PEMs can infer networks accurately and efficiently. Specifically, for slightly autocorrelated time-series data, our approach achieves accuracies higher than or similar to Granger causality, transfer entropy, and convergent crossmapping -- but with much shorter computation time than possible with any of these methods. Our fast and accurate PEMs are easy-to-implement methods for network inference with a clear theoretical underpinning. They provide promising alternatives to current paradigms for the inference of linear models from time-series data, including Granger causality, vector-autoregression, and sparse inverse covariance estimation.

Artificial neural networks simplify complex biological circuits into tractable models for computational exploration and experimentation. However, the simplification of artificial models also undermines their applicability to real brain dynamics. Typical efforts to address this mismatch add complexity to increasingly unwieldy models. Here, we take a different approach; by reducing the complexity of a biological cortical culture, we aim to distil the essential factors of neuronal dynamics and plasticity. We leverage recent advances in growing neurons from human induced pluripotent stem cells (hiPSCs) to analyse ex vivo cortical cultures with only two distinct excitatory and inhibitory neuron populations. Over 6 weeks of development, we record from thousands of neurons using high-density microelectrode arrays (HD-MEAs) that allow access to individual neurons and the broader population dynamics. We compare these dynamics to two-population artificial networks of single-compartment neurons with random sparse connections and show that they produce similar dynamics. Specifically, our model captures the firing and bursting statistics of the cultures. Moreover, tightly integrating models and cultures allows us to evaluate the impact of changing architectures over weeks of development, with and without external stimuli. Broadly, the use of simplified cortical cultures enables us to use the repertoire of theoretical neuroscience techniques established over the past decades on artificial network models. Our approach of deriving neural networks from human cells also allows us, for the first time, to directly compare neural dynamics of disease and control. We found that cultures e.g. from epilepsy patients tended to have increasingly more avalanches of synchronous activity over weeks of development, in contrast to the control cultures. Next, we will test possible interventions, in silico and in vitro, in a drive for personalised approaches to medical care. This work starts bridging an important theoretical-experimental neuroscience gap for advancing our understanding of mammalian neuron dynamics.

Memory consolidation is the process by which recent experiences are assimilated into long-term memory. In animals, this process requires the offline replay of sequences observed during online exploration in the hippocampus. Recent experimental work has found that salient but task-irrelevant stimuli are systematically excluded from these replay epochs, suggesting that replay samples from an abstracted model of the world, rather than verbatim previous experiences. We find that this phenomenon can be explained parsimoniously and biologically plausibly by a Hebbian spike time-dependent plasticity rule at inhibitory synapses. Using spiking networks at three levels of abstraction–leaky integrate-and-fire, biophysically detailed, and abstract binary–we show that this rule enables efficient inference of a model of the structure of the world. While plasticity has previously mainly been studied at excitatory synapses, we find that plasticity at excitatory synapses alone is insufficient to accomplish this type of structural learning. We present theoretical results in a simplified model showing that in the presence of Hebbian excitatory and inhibitory plasticity, the replayed sequences form a statistical estimator of a latent sequence, which converges asymptotically to the ground truth. Our work outlines a direct link between the synaptic and cognitive levels of memory consolidation, and highlights a potential conceptually distinct role for inhibition in computing with SNNs.

The spike-threshold nonlinearity is a fundamental, yet enigmatic, component of biological computation — despite its role in many theories, it has evaded definitive characterisation. Indeed, much classic work has attempted to limit the focus on spiking by smoothing over the spike threshold or by approximating spiking dynamics with firing-rate dynamics. Here, we take a novel perspective that captures the full potential of spike-based computation. Based on previous studies of the geometry of efficient spike-coding networks, we consider a population of neurons with low-rank connectivity, allowing us to cast each neuron’s threshold as a boundary in a space of population modes, or latent variables. Each neuron divides this latent space into subthreshold and suprathreshold areas. We then demonstrate how a network of inhibitory (I) neurons forms a convex, attracting boundary in the latent coding space, and a network of excitatory (E) neurons forms a concave, repellant boundary. Finally, we show how the combination of the two yields stable dynamics at the crossing of the E and I boundaries, and can be mapped onto a constrained optimization problem. The resultant EI networks are balanced, inhibition-stabilized, and exhibit asynchronous irregular activity, thereby closely resembling cortical networks of the brain. Moreover, we demonstrate how such networks can be tuned to either suppress or amplify noise, and how the composition of inhibitory convex and excitatory concave boundaries can result in universal function approximation. Our work puts forth a new theory of biologically-plausible computation in balanced spiking networks, and could serve as a novel framework for scalable and interpretable computation with spikes.

The brain efficiently performs nonlinear computations through its intricate networks of spiking neurons, but how this is done remains elusive. While recurrent spiking networks implementing linear computations can be directly derived and easily understood (e.g., in the spike coding network (SCN) framework), the connectivity required for nonlinear computations can be harder to interpret, as they require additional non-linearities (e.g., dendritic or synaptic) weighted through supervised training. Here we extend the SCN framework to directly implement any polynomial dynamical system. This results in networks requiring multiplicative synapses, which we term the multiplicative spike coding network (mSCN). We demonstrate how the required connectivity for several nonlinear dynamical systems can be directly derived and implemented in mSCNs, without training. We also show how to precisely carry out higher-order polynomials with coupled networks that use only pair-wise multiplicative synapses, and provide expected numbers of connections for each synapse type. Overall, our work provides an alternative method for implementing nonlinear computations in spiking neural networks, while keeping all the attractive features of standard SCNs such as robustness, irregular and sparse firing, and interpretable connectivity. Finally, we discuss the biological plausibility of mSCNs, and how the high accuracy and robustness of the approach may be of interest for neuromorphic computing.

Drosophila olfactory sensory hairs ("sensilla") typically house two olfactory receptor neurons (ORNs) which can laterally inhibit each other via electrical ("ephaptic") coupling. ORN pairing is highly stereotyped and genetically determined. Thus, olfactory signals arriving in the Antennal Lobe (AL) have been pre-processed by a fixed and shallow network at the periphery. To uncover the functional significance of this organization, we developed a nonlinear phenomenological model of asymmetrically coupled ORNs responding to odor mixture stimuli. We derived an analytical solution to the ORNs’ dynamics, which shows that the peripheral network can extract the valence of specific odor mixtures via transient amplification. Our model predicts that for efficient read-out of the amplified valence signal there must exist specific patterns of downstream connectivity that reflect the organization at the periphery. Analysis of AL→Lateral Horn (LH) fly connectomic data reveals evidence directly supporting this prediction. We further studied the effect of ephaptic coupling on olfactory processing in the AL→Mushroom Body (MB) pathway. We show that stereotyped ephaptic interactions between ORNs lead to a clustered odor representation of glomerular responses. Such clustering in the AL is an essential assumption of theoretical studies on odor recognition in the MB. Together our work shows that preprocessing of olfactory stimuli by a fixed and shallow network increases sensitivity to specific odor mixtures, and aids in the learning of novel olfactory stimuli. Work led by Palka Puri, in collaboration with Chih-Ying Su and Shiuan-Tze Wu.

Neural computations are currently investigated using two separate approaches: sorting neurons into functional subpopulations or examining the low-dimensional dynamics of collective activity. Whether and how these two aspects interact to shape computations is currently unclear. Using a novel approach to extract computational mechanisms from networks trained on neuroscience tasks, here we show that the dimensionality of the dynamics and subpopulation structure play fundamentally com- plementary roles. Although various tasks can be implemented by increasing the dimensionality in networks with fully random population structure, flexible input–output mappings instead require a non-random population structure that can be described in terms of multiple subpopulations. Our analyses revealed that such a subpopulation structure enables flexible computations through a mechanism based on gain-controlled modulations that flexibly shape the collective dynamics. Our results lead to task-specific predictions for the structure of neural selectivity, for inactivation experiments and for the implication of different neurons in multi-tasking.

A long standing challenge in biological and artificial intelligence is to understand how new knowledge can be constructed from known building blocks in a way that is amenable for computation by neuronal circuits. Here we focus on the task of storage and recall of structured knowledge in long-term memory. Specifically, we ask how recurrent neuronal networks can store and retrieve multiple knowledge structures. We model each structure as a set of binary relations between events and attributes (attributes may represent e.g., temporal order, spatial location, role in semantic structure), and map each structure to a distributed neuronal activity pattern using a vector symbolic architecture (VSA) scheme. We then use associative memory plasticity rules to store the binarized patterns as fixed points in a recurrent network. By a combination of signal-to-noise analysis and numerical simulations, we demonstrate that our model allows for efficient storage of these knowledge structures, such that the memorized structures as well as their individual building blocks (e.g., events and attributes) can be subsequently retrieved from partial retrieving cues. We show that long-term memory of structured knowledge relies on a new principle of computation beyond the memory basins. Finally, we show that our model can be extended to store sequences of memories as single attractors.

How seizures emerge from the abnormal dynamics of neural networks within the epileptogenic tissue remains an enigma. Are seizures random events, or do detectable changes in brain dynamics precede them? Are mechanisms of seizure emergence identical at the onset and later stages of epilepsy? Is the risk of seizure occurrence stable, or does it change over time? A myriad of questions about seizure genesis remains to be answered to understand the core principles governing seizure genesis. The last decade has brought unprecedented insights into the complex nature of seizure emergence. It is now believed that seizure onset represents the product of the interactions between the process of a transition to seizure, long-term fluctuations in seizure susceptibility, epileptogenesis, and disease progression. During the lecture, we will review the latest observations about mechanisms of ictogenesis operating at multiple temporal scales. We will show how the latest observations contribute to the formation of a comprehensive theory of seizure genesis, and challenge the traditional perspectives on ictogenesis. Finally, we will discuss how combining conventional approaches with computational modeling, modern techniques of in vivo imaging, and genetic manipulation open prospects for exploration of yet hidden mechanisms of seizure genesis.

Resting-state functional magnetic resonance imaging (MRI) has yielded seemingly disparate insights into large-scale organization of the human brain. The brain’s large-scale organization can be divided into two broad categories: zero-lag representations of functional connectivity structure and time-lag representations of traveling wave or propagation structure. In this study, we sought to unify observed phenomena across these two categories in the form of three low-frequency spatiotemporal patterns composed of a mixture of standing and traveling wave dynamics. We showed that a range of empirical phenomena, including functional connectivity gradients, the task-positive/task-negative anti-correlation pattern, the global signal, time-lag propagation patterns, the quasiperiodic pattern and the functional connectome network structure, are manifestations of these three spatiotemporal patterns. These patterns account for much of the global spatial structure that underlies functional connectivity analyses and unifies phenomena in resting-state functional MRI previously thought distinct.

Electrophysiological recordings during perceptual decision tasks in monkeys suggest that the degree of confidence in a decision is based on a simple neural signal produced by the neural decision process. Attractor neural networks provide an appropriate biophysical modeling framework, and account for the experimental results very well. However, it remains unclear whether attractor neural networks can account for confidence reports in humans. We present the results from an experiment in which participants are asked to perform an orientation discrimination task, followed by a confidence judgment. Here we show that an attractor neural network model quantitatively reproduces, for each participant, the relations between accuracy, response times and confidence. We show that the attractor neural network also accounts for confidence-specific sequential effects observed in the experiment (participants are faster on trials following high confidence trials), as well as non confidence-specific sequential effects. Remarkably, this is obtained as an inevitable outcome of the network dynamics, without any feedback specific to the previous decision (that would result in, e.g., a change in the model parameters before the onset of the next trial). Our results thus suggest that a metacognitive process such as confidence in one’s decision is linked to the intrinsically nonlinear dynamics of the decision-making neural network.