Mathematical Modeling seminars

When interacting with complex environments, humans can rapidly adapt their behavior to changes in task or context. To facilitate this adaptation, we often spend substantial periods of time contemplating possible futures before acting. For such planning to be rational, the benefits of planning to future behavior must at least compensate for the time spent thinking. Here we capture these features of human behavior by developing a neural network model where not only actions, but also planning, are controlled by prefrontal cortex. This model consists of a meta-reinforcement learning agent augmented with the ability to plan by sampling imagined action sequences drawn from its own policy, which we refer to as 'rollouts'. Our results demonstrate that this agent learns to plan when planning is beneficial, explaining the empirical variability in human thinking times. Additionally, the patterns of policy rollouts employed by the artificial agent closely resemble patterns of rodent hippocampal replays recently recorded in a spatial navigation task, in terms of both their spatial statistics and their relationship to subsequent behavior. Our work provides a new theory of how the brain could implement planning through prefrontal-hippocampal interactions, where hippocampal replays are triggered by - and in turn adaptively affect - prefrontal dynamics.

Critical phenomena abound in nature, from forest fires and earthquakes to avalanches in sand and neuronal activity. Since the 2003 publication by Beggs & Plenz on neuronal avalanches, a growing body of work suggests that the brain homeostatically regulates itself to operate near a critical point where information processing is optimal. At this critical point, incoming activity is neither amplified (supercritical) nor damped (subcritical), but approximately preserved as it passes through neural networks. Departures from the critical point have been associated with conditions of poor neurological health like epilepsy, Alzheimer's disease, and depression. One complication that arises from this picture is that the critical point assumes no external input. But, biological neural networks are constantly bombarded by external input. How is then the brain able to homeostatically adapt near the critical point? We’ll see that the theory of quasicriticality, an organizing principle for brain dynamics, can account for this paradoxical situation. As external stimuli drive the cortex, quasicriticality predicts a departure from criticality while maintaining optimal properties for information transmission. We’ll see that simulations and experimental data confirm these predictions and describe new ones that could be tested soon. More importantly, we will see how this organizing principle could help in the search for biomarkers that could soon be tested in clinical studies.

The critical brain hypothesis states that the brain can benefit from operating close to a second-order phase transition. While it has been shown that several computational aspects of sensory information processing (e.g., sensitivity to input) are optimal in this regime, it is still unclear whether these computational benefits of criticality can be leveraged by neural systems performing behaviorally relevant computations. To address this question, we investigate signatures of criticality in networks optimized to perform efficient encoding. We consider a network of leaky integrate-and-fire neurons with synaptic transmission delays and input noise. Previously, it was shown that the performance of such networks varies non-monotonically with the noise amplitude. Interestingly, we find that in the vicinity of the optimal noise level for efficient coding, the network dynamics exhibits signatures of criticality, namely, the distribution of avalanche sizes follows a power law. When the noise amplitude is too low or too high for efficient coding, the network appears either super-critical or sub-critical, respectively. This result suggests that two influential, and previously disparate theories of neural processing optimization—efficient coding, and criticality—may be intimately related

Neural activity is often described in terms of population-level factors extracted from the responses of many neurons. Factors provide a lower-dimensional description with the aim of shedding light on network computations. Yet, mechanistically, computations are performed not by continuously valued factors but by interactions among neurons that spike discretely and variably. Models provide a means of bridging these levels of description. We developed a general method for training model networks of spiking neurons by leveraging factors extracted from either data or firing-rate-based networks. In addition to providing a useful model-building framework, this formalism illustrates how reliable and continuously valued factors can arise from seemingly stochastic spiking. Our framework establishes procedures for embedding this property in network models with different levels of realism. The relationship between spikes and factors in such networks provides a foundation for interpreting (and subtly redefining) commonly used quantities such as firing rates.

The eye is a multi-component biological system, where mechanics, optics, transport phenomena and chemical reactions are strictly interlaced, characterized by the typical bio-variability in sizes and material properties. The eye’s response to external action is patient-specific and it can be predicted only by a customized approach, that accounts for the multiple physics and for the intrinsic microstructure of the tissues, developed with the aid of forefront means of computational biomechanics. Our activity in the last years has been devoted to the development of a comprehensive model of the cornea that aims at being entirely patient-specific. While the geometrical aspects are fully under control, given the sophisticated diagnostic machinery able to provide a fully three-dimensional images of the eye, the major difficulties are related to the characterization of the tissues, which require the setup of in-vivo tests to complement the well documented results of in-vitro tests. The interpretation of in-vivo tests is very complex, since the entire structure of the eye is involved and the characterization of the single tissue is not trivial. The availability of micromechanical models constructed from detailed images of the eye represents an important support for the characterization of the corneal tissues, especially in the case of pathologic conditions. In this presentation I will provide an overview of the research developed in our group in terms of computational models and experimental approaches developed for the human cornea.

Humans constantly make perceptual decisions on human faces and voices. These regularly come with the challenge of receiving only uncertain sensory evidence, resulting from noisy input and noisy neural processes. Efficiently adapting one’s internal decision system including prior expectations and subsequent metacognitive assessments to these challenges is crucial in everyday life. However, the exact decision mechanisms and whether these represent modifiable states remain unknown in the general population and clinical patients with psychosis. Using data from a laboratory-based sample of healthy controls and patients with psychosis as well as a complementary, large online sample of healthy controls, I will demonstrate how a combination of perceptual face and voice recognition decision fidelity, metacognitive ratings, and Bayesian computational modelling may be used as indicators to differentiate between non-clinical and clinical states in the future.

Learning in neural networks requires assigning the right values to thousands to trillions or more of individual connections, so that the network as a whole produces the desired behavior. Neuroscientists have gained insights into this “credit assignment” problem through decades of experimental, modeling, and theoretical studies. This has suggested key roles for synaptic eligibility traces and top-down feedback signals, among other factors. Here we study the potential contribution of another type of signaling that is being revealed in greater and greater fidelity by ongoing molecular and genomics studies. This is the set of modulatory pathways local to a given circuit, which form an intriguing second type of connectome overlayed on top of synaptic connectivity. We will share ongoing modeling and theoretical work that explores the possible roles of this local modulatory connectome in network learning.

What is the relationship between task, network architecture, and population activity in nonlinear deep networks? I will describe the Gated Deep Linear Network framework, which schematizes how pathways of information flow impact learning dynamics within an architecture. Because of the gating, these networks can compute nonlinear functions of their input. We derive an exact reduction and, for certain cases, exact solutions to the dynamics of learning. The reduction takes the form of a neural race with an implicit bias towards shared representations, which then govern the model’s ability to systematically generalize, multi-task, and transfer. We show how appropriate network architectures can help factorize and abstract knowledge. Together, these results begin to shed light on the links between architecture, learning dynamics and network performance.

Humans and animals learn and plan with flexibility and efficiency well beyond that of modern Machine Learning methods. This is hypothesized to owe in part to the ability of animals to build structured representations of their environments, and modulate these representations to rapidly adapt to new settings. In the first part of this talk, I will discuss theoretical work describing how learned representations in hippocampus enable rapid adaptation to new goals by learning predictive representations, while entorhinal cortex compresses these predictive representations with spectral methods that support smooth generalization among related states. I will also cover recent work extending this account, in which we show how the predictive model can be adapted to the probabilistic setting to describe a broader array of generalization results in humans and animals, and how entorhinal representations can be modulated to support sample generation optimized for different behavioral states. In the second part of the talk, I will overview some of the ways in which we have combined many of the same mathematical concepts with state-of-the-art deep learning methods to improve efficiency and performance in machine learning applications like physical simulation, relational reasoning, and design.

It is widely accepted that human cognition is the product of spiking neurons. Yet even for basic cognitive functions, such as the ability to make decisions or prepare and execute a voluntary movement, the gap between spikes and computation is vast. Only for very simple circuits and reflexes can one explain computations neuron-by-neuron and spike-by-spike. This approach becomes infeasible when neurons are numerous the flow of information is recurrent. To understand computation, one thus requires appropriate abstractions. An increasingly common abstraction is the neural ‘factor’. Factors are central to many explanations in systems neuroscience. Factors provide a framework for describing computational mechanism, and offer a bridge between data and concrete models. Yet there remains some discomfort with this abstraction, and with any attempt to provide mechanistic explanations above that of spikes, neurons, cell-types, and other comfortingly concrete entities. I will explain why, for many networks of spiking neurons, factors are not only a well-defined abstraction, but are critical to understanding computation mechanistically. Indeed, factors are as real as other abstractions we now accept: pressure, temperature, conductance, and even the action potential itself. I use recent empirical results to illustrate how factor-based hypotheses have become essential to the forming and testing of scientific hypotheses. I will also show how embracing factor-level descriptions affords remarkable power when decoding neural activity for neural engineering purposes.

Computation in neural circuits relies on a common set of motifs, including divergence of common inputs to parallel pathways, convergence of multiple inputs to a single neuron, and nonlinearities that select some signals over others. Convergence and circuit nonlinearities, considered individually, can lead to a loss of information about the inputs. Past work has detailed how to optimize nonlinearities and circuit weights to maximize information, but we show that selective nonlinearities, acting together with divergent and convergent circuit structure, can improve information transmission over a purely linear circuit despite the suboptimality of these components individually. These nonlinearities recode the inputs in a manner that preserves the variance among converged inputs. Our results suggest that neural circuits may be doing better than expected without finely tuned weights.

Modern electrophysiological recordings simultaneously capture single-unit spiking activities of hundreds of neurons. These neurons exhibit highly complex coordination patterns. Where does this complexity stem from? One candidate is the ubiquitous heterogeneity in connectivity of local neural circuits. Studying neural network dynamics in the linearized regime and using tools from statistical field theory of disordered systems, we derive relations between structure and dynamics that are readily applicable to subsampled recordings of neural circuits: Measuring the statistics of pairwise covariances allows us to infer statistical properties of the underlying connectivity. Applying our results to spontaneous activity of macaque motor cortex, we find that the underlying network operates in a strongly recurrent regime. In this regime, network connectivity is highly heterogeneous, as quantified by a large radius of bulk connectivity eigenvalues. Being close to the point of linear instability, this dynamical regime predicts a rich correlation structure, a large dynamical repertoire, long-range interaction patterns, relatively low dimensionality and a sensitive control of neuronal coordination. These predictions are verified in analyses of spontaneous activity of macaque motor cortex and mouse visual cortex. Finally, we show that even microscopic features of connectivity, such as connection motifs, systematically scale up to determine the global organization of activity in neural circuits.

Explicit similarity judgments tend to emphasize relational information more than do difference judgments. In this talk, I propose and test the hypothesis that this asymmetry arises because human reasoners represent the relation different as the negation of the relation same (i.e., as not-same). This proposal implies that processing difference is more cognitively demanding than processing similarity. Both for verbal comparisons between word pairs, and for visual comparisons between sets of geometric shapes, participants completed a triad task in which they selected which of two options was either more similar to or more different from a standard. On unambiguous trials, one option was unambiguously more similar to the standard, either by virtue of featural similarity or by virtue of relational similarity. On ambiguous trials, one option was more featurally similar (but less relationally similar) to the standard, whereas the other was more relationally similar (but less featurally similar). Given the higher cognitive complexity of assessing relational similarity, we predicted that detecting relational difference would be particularly demanding. We found that participants (1) had more difficulty accurately detecting relational difference than they did relational similarity on unambiguous trials, and (2) tended to emphasize relational information more when judging similarity than when judging difference on ambiguous trials. The latter finding was captured by a computational model of comparison that weights relational information more heavily for similarity than for difference judgments. These results provide convergent evidence for a representational asymmetry between the relations same and different.

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.

During navigation, animals estimate their position using path integration and landmarks. In a series of two studies, we used virtual reality and electrophysiology to dissect how these inputs combine to generate the brain’s spatial representations. In the first study (Campbell et al., 2018), we focused on the medial entorhinal cortex (MEC) and its set of navigationally-relevant cell types, including grid cells, border cells, and speed cells. We discovered that attractor dynamics could explain an array of initially puzzling MEC responses to virtual reality manipulations. This theoretical framework successfully predicted both MEC grid cell responses to additional virtual reality manipulations, as well as mouse behavior in a virtual path integration task. In the second study (Campbell*, Attinger* et al., 2021), we asked whether these principles generalize to other navigationally-relevant brain regions. We used Neuropixels probes to record thousands of neurons from MEC, primary visual cortex (V1), and retrosplenial cortex (RSC). In contrast to the prevailing view that “everything is everywhere all at once,” we identified a unique population of MEC neurons, overlapping with grid cells, that became active with striking spatial periodicity while head-fixed mice ran on a treadmill in darkness. These neurons exhibited unique cue-integration properties compared to other MEC, V1, or RSC neurons: they remapped more readily in response to conflicts between path integration and landmarks; they coded position prospectively as opposed to retrospectively; they upweighted path integration relative to landmarks in conditions of low visual contrast; and as a population, they exhibited a lower-dimensional activity structure. Based on these results, our current view is that MEC attractor dynamics play a privileged role in resolving conflicts between path integration and landmarks during navigation. Future work should include carefully designed causal manipulations to rigorously test this idea, and expand the theoretical framework to incorporate notions of uncertainty and optimality.

Cognitive science and cognitive neuroscience researchers have agreed that the replication of findings is important for establishing which ideas (or theories) are integral to the study of cognition across the lifespan. Recently, high-profile papers have called into question findings that were once thought to be unassailable. Much attention has been paid to how p-hacking, publication bias, and sample size are responsible for failed replications. However, much less attention has been paid to the fidelity by which researchers enact study protocols. Researchers conducting education or clinical trials are aware of the importance in fidelity – or the extent to which the protocols are delivered in the same way across participants. Nevertheless, this idea has not been applied to cognitive contexts. This seminar discusses factors that impact the replicability of findings alongside recent models suggesting that even small fidelity deviations have real impacts on the data collected.

Neurons in the primary visual cortex (V1) of rodents are selective to the orientation of the stimulus, as in other mammals such as cats and monkeys. However, in contrast with those species, their neurons display a very different type of spatial organization. Instead of orientation maps they are organized in a “salt and pepper” pattern, where adjacent neurons have completely different preferred orientations. This structure has motivated both experimental and theoretical research with the objective of determining which aspects of the connectivity patterns and intrinsic neuronal responses can explain the observed behavior. These analysis have to take into account also that the neurons of the thalamus that send their outputs to the cortex have more complex responses in rodents than in higher mammals, displaying, for instance, a significant degree of orientation selectivity. In this talk we present work showing that a random feed-forward connectivity pattern, in which the probability of having a connection between a cortical neuron and a thalamic neuron depends only on the relative distance between them is enough explain several aspects of the complex phenomenology found in these systems. Moreover, this approach allows us to evaluate analytically the statistical structure of the thalamic input on the cortex. We find that V1 neurons are orientation selective but the preferred orientation of the stimulus depends on the spatial frequency of the stimulus. We disentangle the effect of the non circular thalamic receptive fields, finding that they control the selectivity of the time-averaged thalamic input, but not the selectivity of the time locked component. We also compare with experiments that use reverse correlation techniques, showing that ON and OFF components of the aggregate thalamic input are spatially segregated in the cortex.

The affinity between statistical physics and machine learning has a long history. I will describe the main lines of this long-lasting friendship in the context of current theoretical challenges and open questions about deep learning. Theoretical physics often proceeds in terms of solvable synthetic models, I will describe the related line of work on solvable models of simple feed-forward neural networks. I will highlight a path forward to capture the subtle interplay between the structure of the data, the architecture of the network, and the optimization algorithms commonly used for learning.

Like all neural tissue, the retina has a high metabolic demand, and requires a constant supply of oxygen. Second and third order neurons are supplied by the retinal circulation, whose characteristics are similar to brain circulation. However, the photoreceptor region, which occupies half of the retinal thickness, is avascular, and relies on diffusion of oxygen from the choroidal circulation, whose properties are very different, as well as the retinal circulation. By fitting diffusion models to oxygen measurements made with oxygen microelectrodes, it is possible to understand the relative roles of the two circulations under normal conditions of light and darkness, and what happens if the retina is detached or the retinal circulation is occluded. Most of this work has been done in vivo in rat, cat, and monkey, but recent work in the isolated mouse retina will also be discussed.