Dynamical Systems seminars

Efficient navigation requires animals to track their position, velocity and heading direction (HD). Some animals’ behavior suggests that they also track uncertainties about these navigational variables, and make strategic use of these uncertainties, in line with a Bayesian computation. Ring-attractor networks have been proposed to estimate and track these navigational variables, for instance in the HD system of the fruit fly Drosophila. However, such networks are not designed to incorporate a notion of uncertainty, and therefore seem unsuited to implement dynamic Bayesian inference. Here, we close this gap by showing that specifically tuned ring-attractor networks can track both a HD estimate and its associated uncertainty, thereby approximating a circular Kalman filter. We identified the network motifs required to integrate angular velocity observations, e.g., through self-initiated turns, and absolute HD observations, e.g., visual landmark inputs, according to their respective reliabilities, and show that these network motifs are present in the connectome of the Drosophila HD system. Specifically, our network encodes uncertainty in the amplitude of a localized bump of neural activity, thereby generalizing standard ring attractor models. In contrast to such standard attractors, however, proper Bayesian inference requires the network dynamics to operate in a regime away from the attractor state. More generally, we show that near-Bayesian integration is inherent in generic ring attractor networks, and that their amplitude dynamics can account for close-to-optimal reliability weighting of external evidence for a wide range of network parameters. This only holds, however, if their connection strengths allow the network to sufficiently deviate from the attractor state. Overall, our work offers a novel interpretation of ring attractor networks as implementing dynamic Bayesian integrators. We further provide a principled theoretical foundation for the suggestion that the Drosophila HD system may implement Bayesian HD tracking via ring attractor dynamics.

Animals exhibit remarkable learning capabilities with little direct supervision. Likewise, self-supervised learning is an emergent paradigm in artificial intelligence, closing the performance gap to supervised learning. In the context of biology, self-supervised learning corresponds to a setting where one sense or specific stimulus may serve as a supervisory signal for another. After learning, the latter can be used to predict the former. On the implementation level, it has been demonstrated that such predictive learning can occur at the single neuron level, in compartmentalized neurons that separate and associate information from different streams. We demonstrate the power such self-supervised learning over unsupervised (Hebb-like) learning rules, which depend heavily on stimulus statistics, in two examples: First, in the context of animal navigation where predictive learning can associate internal self-motion information always available to the animal with external visual landmark information, leading to accurate path-integration in the dark. We focus on the well-characterized fly head direction system and show that our setting learns a connectivity strikingly similar to the one reported in experiments. The mature network is a quasi-continuous attractor and reproduces key experiments in which optogenetic stimulation controls the internal representation of heading, and where the network remaps to integrate with different gains. Second, we show that incorporating global gating by reward prediction errors allows the same setting to learn conditioning at the neuronal level with mixed selectivity. At its core, conditioning entails associating a neural activity pattern induced by an unconditioned stimulus (US) with the pattern arising in response to a conditioned stimulus (CS). Solving the generic problem of pattern-to-pattern associations naturally leads to emergent cognitive phenomena like blocking, overshadowing, saliency effects, extinction, interstimulus interval effects etc. Surprisingly, we find that the same network offers a reductionist mechanism for causal inference by resolving the post hoc, ergo propter hoc fallacy.

Neuronal network computation and computation by avalanche supporting networks are of interest to the fields of physics, computer science (computation theory as well as statistical or machine learning) and neuroscience. Here we show that computation of complex Boolean functions arises spontaneously in threshold networks as a function of connectivity and antagonism (inhibition), computed by logic automata (motifs) in the form of computational cascades. We explain the emergent inverse relationship between the computational complexity of the motifs and their rank-ordering by function probabilities due to motifs, and its relationship to symmetry in function space. We also show that the optimal fraction of inhibition observed here supports results in computational neuroscience, relating to optimal information processing.

Spiking recurrent neural networks (RNNs) are a promising tool for solving a wide variety of complex cognitive and motor tasks, due to their rich temporal dynamics and sparse processing. However training spiking RNNs on dedicated neuromorphic hardware is still an open challenge. This is due mainly to the lack of local, hardware-friendly learning mechanisms that can solve the temporal credit assignment problem and ensure stable network dynamics, even when the weight resolution is limited. These challenges are further accentuated, if one resorts to using memristive devices for in-memory computing to resolve the von-Neumann bottleneck problem, at the expense of a substantial increase in variability in both the computation and the working memory of the spiking RNNs. In this talk, I will present our recent work where we introduced a PyTorch simulation framework of memristive crossbar arrays that enables accurate investigation of such challenges. I will show that recently proposed e-prop learning rule can be used to train spiking RNNs whose weights are emulated in the presented simulation framework. Although e-prop locally approximates the ideal synaptic updates, it is difficult to implement the updates on the memristive substrate due to substantial device non-idealities. I will mention several widely adapted weight update schemes that primarily aim to cope with these device non-idealities and demonstrate that accumulating gradients can enable online and efficient training of spiking RNN on memristive substrates.

Fate decisions in developing tissues involve cells transitioning between a set of discrete cell states. Geometric models, often referred to as Waddington landscapes, are an appealing way to describe differentiation dynamics and developmental decisions. We consider the differentiation of neural and mesodermal cells from pluripotent mouse embryonic stem cells exposed to different combinations and durations of signalling factors. We developed a principled statistical approach using flow cytometry data to quantify differentiating cell states. Then, using a framework based on Catastrophe Theory and approximate Bayesian computation, we constructed the corresponding dynamical landscape. The result was a quantitative model that accurately predicted the proportions of neural and mesodermal cells differentiating in response to specific signalling regimes. Taken together, the approach we describe is broadly applicable for the quantitative analysis of differentiation dynamics and for determining the logic of developmental cell fate decisions.

Change is ubiquitous in living beings. In particular, the connectome and neural representations can change. Nevertheless behaviors and memories often persist over long times. In a standard model, associative memories are represented by assemblies of strongly interconnected neurons. For faithful storage these assemblies are assumed to consist of the same neurons over time. We propose a contrasting memory model with complete temporal remodeling of assemblies, based on experimentally observed changes of synapses and neural representations. The assemblies drift freely as noisy autonomous network activity or spontaneous synaptic turnover induce neuron exchange. The exchange can be described analytically by reduced, random walk models derived from spiking neural network dynamics or from first principles. The gradual exchange allows activity-dependent and homeostatic plasticity to conserve the representational structure and keep inputs, outputs and assemblies consistent. This leads to persistent memory. Our findings explain recent experimental results on temporal evolution of fear memory representations and suggest that memory systems need to be understood in their completeness as individual parts may constantly change.

The free-energy principle and active inference have received a significant attention in the fields of neuroscience and machine learning. However, it remains to be established whether active inference is an apt explanation for any given neural network that actively exchanges with its environment. To address this issue, we show that a class of canonical neural networks of rate coding models implicitly performs variational Bayesian inference under a well-known form of partially observed Markov decision process model (Isomura, Shimazaki, Friston, Commun Biol, 2022). Based on the proposed theory, we demonstrate that canonical neural networks—featuring delayed modulation of Hebbian plasticity—can perform planning and adaptive behavioural control in the Bayes optimal manner, through postdiction of their previous decisions. This scheme enables us to estimate implicit priors under which the agent’s neural network operates and identify a specific form of the generative model. The proposed equivalence is crucial for rendering brain activity explainable to better understand basic neuropsychology and psychiatric disorders. Moreover, this notion can dramatically reduce the complexity of designing self-learning neuromorphic hardware to perform various types of tasks.

Living organisms have mastered the dynamic control of stresses within sheets to induce shape transformation and locomotion. For instance, the spatiotemporal pattern of action potential in a heart yields a dynamical stress field leading to shape changes and biological function. Such structures inspired the development of theoretical tools and responsive materials alike. Yet, present attempts to mimic their rich dynamics and phenomenology in autonomous synthetic matter are still very limited. In this talk, I will present several complementing innovations toward this goal: novel shaping mechanisms that were overlooked by previous research, new fabrication techniques for programmable matter via 4D printing of gel structures, and most prominently, the first autonomous shape morphing membranes. The dynamical control over the geometry of the material is a prevalent theme in all of these achievements. In particular, the latter system demonstrates localized deformations, induced by a pattern-forming chemical reaction, that prescribe the patterns of curvature, leading to global shape evolution. Together, these developments present a route for modeling and producing fully autonomous soft membranes mimicking some of the locomotive capabilities of living organisms.

Many engineered and biological systems must trade off performance and energy use, and the brain is no exception. While there are theories on how activity levels are controlled in biological networks through feedback control (homeostasis), it is not clear what the effects on population coding are, and therefore how performance and energy can be traded off. In this talk we will consider this tradeoff in auto-encoding networks, in which there is a clear definition of performance (the coding loss). We first show how SNNs follow a characteristic trade-off curve between activity levels and coding loss, but that standard networks need to be retrained to achieve different tradeoff points. We next formalize this tradeoff with a joint loss function incorporating coding loss (performance) and activity loss (energy use). From this loss we derive a class of spiking networks which coordinates its spiking to minimize both the activity and coding losses -- and as a result can dynamically adjust its coding precision and energy use. The network utilizes several known activity control mechanisms for this --- threshold adaptation and feedback inhibition --- and elucidates their potential function within neural circuits. Using geometric intuition, we demonstrate how these mechanisms regulate coding precision, and thereby performance. Lastly, we consider how these insights could be transferred to trained SNNs. Overall, this work addresses a key energy-coding trade-off which is often overlooked in network studies, expands on our understanding of homeostasis in biological SNNs, as well as provides a clear framework for considering performance and energy use in artificial SNNs.

The ability to discriminate different sensory patterns by disentangling their neural representations is an important property of neural networks. While a variety of learning rules are known to be highly effective at fine-tuning synapses to achieve this, less is known about how different cell types in the brain can facilitate this process by providing architectural priors that bias the network towards sparse, selective, and discriminable representations. We studied this by simulating a neuronal network modelled on the dentate gyrus—an area characterised by sparse activity associated with pattern separation in spatial memory tasks. To test the contribution of different cell types to these functions, we presented the model with a wide dynamic range of input patterns and systematically added or removed different circuit elements. We found that recruiting feedback inhibition indirectly via recurrent excitatory neurons proved particularly helpful in disentangling patterns, and show that simple alignment principles for excitatory and inhibitory connections are a highly effective strategy.

Reservoir computing is a promising framework to study cortical computation, as it is based on continuous, online processing and the requirements and operating principles are compatible with cortical circuit dynamics. However, the framework has issues that limit its scope as a generic model for cortical processing. The most obvious of these is that, in traditional models, learning is restricted to the output projections and takes place in a fully supervised manner. If such an output layer is interpreted at face value as downstream computation, this is biologically questionable. If it is interpreted merely as a demonstration that the network can accurately represent the information, this immediately raises the question of what would be biologically plausible mechanisms for transmitting the information represented by a reservoir and incorporating it in downstream computations. Another major issue is that we have as yet only modest insight into how the structural and dynamical features of a network influence its computational capacity, which is necessary not only for gaining an understanding of those features in biological brains, but also for exploiting reservoir computing as a neuromorphic application. In this talk, I will first demonstrate a method for quantifying the representational capacity of reservoirs without training them on tasks. Based on this technique, which allows systematic comparison of systems, I then present our recent work towards understanding the roles of heterogeneity and connectivity patterns in enhancing both the computational properties of a network and its ability to reliably transmit to downstream networks. Finally, I will give a brief taster of our current efforts to apply the reservoir computing framework to magnetic systems as an approach to neuromorphic computing.

Intelligent behavior depends on the brain’s ability to anticipate future events. However, the learning rules that enable neurons to predict and fire ahead of sensory inputs remain largely unknown. We propose a plasticity rule based on pre-dictive processing, where the neuron learns a low-rank model of the synaptic input dynamics in its membrane potential. Neurons thereby amplify those synapses that maximally predict other synaptic inputs based on their temporal relations, which provide a solution to an optimization problem that can be implemented at the single-neuron level using only local information. Consequently, neurons learn sequences over long timescales and shift their spikes towards the first inputs in a sequence. We show that this mechanism can explain the development of anticipatory motion signaling and recall in the visual system. Furthermore, we demonstrate that the learning rule gives rise to several experimentally observed STDP (spike-timing-dependent plasticity) mechanisms. These findings suggest prediction as a guiding principle to orchestrate learning and synaptic plasticity in single neurons.

Timescales characterize how fast the observables change in time. In neuroscience, they can be estimated from the measured activity and can be used, for example, as a signature of the memory trace in the network. I will first discuss the inference of the timescales from the neuroscience data comprised of the short trials and introduce a new unbiased method. Then, I will apply the method to the data recorded from a local population of cortical neurons from the visual area V4. I will demonstrate that the ongoing spiking activity unfolds across at least two distinct timescales - fast and slow - and the slow timescale increases when monkeys attend to the location of the receptive field. Which models can give rise to such behavior? Random balanced networks are known for their fast timescales; thus, a change in the neurons or network properties is required to mimic the data. I will propose a set of models that can control effective timescales and demonstrate that only the model with strong recurrent interactions fits the neural data. Finally, I will discuss the timescales' relevance for behavior and cortical computations.

It was long hypothesized that natural systems might take advantage of the extended temporal and spatial correlations close to the critical point to improve their computational capabilities. However, on the other side, different distances to criticality were inferred from the recordings of nervous systems. In my talk, I discuss how including additional constraints on the processing time can shift the optimal operating point of the recurrent networks. Moreover, the data from the visual cortex of the monkeys during the attentional task indicate that they flexibly change the closeness to the critical point of the local activity. Overall it suggests that, as we would expect from common sense, the optimal state depends on the task at hand, and the brain adapts to it in a local and fast manner.

Excitatory and inhibitory (E & I) inputs to cortical neurons remain balanced across different conditions. The balanced network model provides a self-consistent account of this observation: population rates dynamically adjust to yield a state in which all neurons are active at biological levels, with their E & I inputs tightly balanced. But global tight E/I balance predicts population responses with linear stimulus-dependence and does not account for systematic cortical response nonlinearities such as divisive normalization, a canonical brain computation. However, when necessary connectivity conditions for global balance fail, states arise in which only a localized subset of neurons are active and have balanced inputs. We analytically show that in networks of neurons with different stimulus selectivities, the emergence of such localized balance states robustly leads to normalization, including sublinear integration and winner-take-all behavior. An alternative model that exhibits normalization is the Stabilized Supralinear Network (SSN), which predicts a regime of loose, rather than tight, E/I balance. However, an understanding of the causal relationship between E/I balance and normalization in SSN and conditions under which SSN yields significant sublinear integration are lacking. For weak inputs, SSN integrates inputs supralinearly, while for very strong inputs it approaches a regime of tight balance. We show that when this latter regime is globally balanced, SSN cannot exhibit strong normalization for any input strength; thus, in SSN too, significant normalization requires localized balance. In summary, we causally and quantitatively connect a fundamental feature of cortical dynamics with a canonical brain computation. Time allowing I will also cover our work extending a normative theoretical account of normalization which explains it as an example of efficient coding of natural stimuli. We show that when biological noise is accounted for, this theory makes the same prediction as the SSN: a transition to supralinear integration for weak stimuli.

Resonance is defined as a maximal amplification of the response of a system to periodic inputs in a limited, intermediate input frequency band. Resonance may serve to optimize inter-neuronal communication, and has been observed at multiple levels of neuronal organization including membrane potential fluctuations, single neuron spiking, postsynaptic potentials, and neuronal networks. However, it is unknown how resonance observed at one level of neuronal organization (e.g., network) depends on the properties of the constituting building blocks, and whether, and if yes how, it affects the resonant and oscillatory properties upstream. One difficulty is the absence of a conceptual framework that facilitates the interrogation of resonant neuronal circuits and organizes the mechanistic investigation of network resonance in terms of the circuit components, across levels of organization. We address these issues by discussing a number of representative case studies. The dynamic mechanisms responsible for the generation of resonance involve disparate processes, including negative feedback effects, history-dependence, spiking discretization combined with subthreshold passive dynamics, combinations of these, and resonance inheritance from lower levels of organization. The band-pass filters associated with the observed resonances are generated by primarily nonlinear interactions of low- and high-pass filters. We identify these filters (and interactions) and we argue that these are the constitutive building blocks of a resonance framework. Finally, we discuss alternative frameworks and we show that different types of models (e.g., spiking neural networks and rate models) can show the same type of resonance by qualitative different mechanisms.

Living Systems often seem to follow, in addition to external constraints and interactions, an intrinsic predictive model of the world — a defining trait of Anticipatory Systems. Here we study rhythmic behaviour in Caulerpa, a marine green alga, which appears to predict the day/night light cycle. Caulerpa consists of differentiated organs resembling leaves, stems and roots. While an individual can exceed a meter in size, it is a single multinucleated giant cell. Active transport has been hypothesized to play a key role in organismal development. It has been an open question in the literature whether rhythmic transport phenomena in this organism are of autonomous circadian nature. Using Raspberry-Pi cameras, we track over weeks the morphogenesis of tens of samples concurrently, while tracing at resolution of tens of seconds the variation of the green coverage. The latter reveals waves propagating over centimeters within few hours, and is attributed to chloroplast redistribution at whole-organism scale. Our observations of algal segments regenerating under 12-hour light/dark cycles indicate that the initiation of the waves precedes the external light change. Using time-frequency analysis, we find that the temporal spectrum of these green pulses contains a circadian period. The latter persists over days even under constant illumination, indicative of its autonomous nature. We further explore the system under non-circadian periods, to reveal how the spectral content changes in response. Time-keeping and synchronization are recurring themes in biological research at various levels of description — from subcellular components to ecological systems. We present a seemingly primitive living system that exhibits apparent anticipatory behaviour. This research offers quantitative constraints for theoretical frameworks of such systems.