Mathematical Modeling seminars

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.

Individuals vary widely in how they categorize novel phenomena. This individual variation has led canonical theories in cognitive and social science to suggest that communication in large social networks leads populations to construct divergent category systems. Yet, anthropological data indicates that large, independent societies consistently arrive at similar categories across a range of topics. How is it possible for diverse populations, consisting of individuals with significant variation in how they view the world, to independently construct similar categories? Through a series of online experiments, I show how large communication networks within cultures can promote the formation of similar categories across cultures. For this investigation, I designed an online “Grouping Game” to observe how people construct categories in both small and large populations when tasked with grouping together the same novel and ambiguous images. I replicated this design for English-speaking subjects in the U.S. and Mandarin-speaking subjects in China. In both cultures, solitary individuals and small social groups produced highly divergent category systems. Yet, large social groups separately and consistently arrived at highly similar categories both within and across cultures. These findings are accurately predicted by a simple mathematical model of critical mass dynamics. Altogether, I show how large communication networks can filter lexical diversity among individuals to produce replicable society-level patterns, yielding unexpected implications for cultural evolution. In particular, I discuss how participants in both cultures readily harnessed analogies when categorizing novel stimuli, and I examine the role of communication networks in promoting cross-cultural similarities in analogy-making as the key engine of category formation.

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.

The basal ganglia are a collection of brain areas that are connected by a variety of synaptic pathways and are a site of significant reward-related dopamine release. These properties suggest a possible role for the basal ganglia in action selection, guided by reinforcement learning. In this talk, I will discuss a framework for how this function might be performed and computational results using an upward mapping to identify putative low-dimensional control ensembles that may be involved in tuning decision policy. I will also present some recent experimental results and theory – related to effects of extracellular ion dynamics -- that run counter to the classical view of basal ganglia pathways and suggest a new interpretation of certain aspects of this framework. For those not so interested in the basal ganglia, I hope that the upward mapping approach and impact of extracellular ion dynamics will nonetheless be of interest!

From a workout at the gym to helping a colleague with their work, everyday we make decisions about whether we are willing to exert effort to obtain some sort of benefit. Increases in how effortful actions and cognitive processes are perceived to be has been linked to clinically severe impairments to motivation, such as apathy and fatigue, across many neurological and psychiatric conditions. However, the vast majority of neuroscience research has focused on understanding the benefits for acting, the rewards, and not on the effort required. As a result, the computational and neural mechanisms underlying how effort is processed are poorly understood. How do we compute how effortful we perceive a task to be? How does this feed into our motivation and decisions of whether to act? How are such computations implemented in the brain? and how do they change in different environments? I will present a series of studies examining these questions using novel behavioural tasks, computational modelling, fMRI, pharmacological manipulations, and testing in a range of different populations. These studies highlight how the brain represents the costs of exerting effort, and the dynamic processes underlying how our sensitivity to effort changes as a function of our goals, traits, and socio-cognitive processes. This work provides new computational frameworks for understanding and examining impaired motivation across psychiatric and neurological conditions, as well as why all of us, sometimes, can’t be bothered.

An equation of state is something you hear about in introductory thermodynamics, for example, the Ideal gas equation. The ideal gas equation relates the pressure, volume, and the number of particles of the gas, to its temperature, uniquely defining its state. This description is possible in physics when the system under investigation is in equilibrium or near equilibrium. In biology, a tissue is modeled as a fluid composed of cells. These cells are constantly interacting with each other through mechanical and chemical signaling, driving them far from equilibrium. Can an equation of state exist for such a messy interacting system? In this talk, I show that the presence of strong cell-cell interaction in tissues gives rise to a novel non-equilibrium, size-dependent surface tension, something unheard of for classical fluids. This surface tension, in turn, modifies the packing of cells inside the tissue generating a size-dependent density and pressure. Finally, we show that a combination of these non-equilibrium pressure and densities can yield an equation of state for biological tissues arbitrarily far from equilibrium. In the end, I discuss how this new paradigm of size-dependent biological properties gives rise to novel modes of cellular motion in tissues

Synaptic transmission provides the basis for neuronal communication. When an action-potential propagates through the axonal arbour, it activates voltage-gated Ca2+ channels located in the vicinity of release-ready synaptic vesicles docked at the presynaptic active zone. Ca2+ ions enter the presynaptic terminal and activate the vesicular Ca2+ sensor, thereby triggering neurotransmitter release. This whole process occurs on a timescale of a few milliseconds. In addition to fast, synchronous release, which keeps pace with action potentials, many synapses also exhibit delayed asynchronous release that persists for tens to hundreds of milliseconds. In this talk I will demonstrate how experimentally constrained computational modelling of underlying biological processes can complement laboratory studies (using electrophysiology and imaging techniques) and provide insights into the mechanisms of synaptic transmission.

One elementary brain function that underlies many of our cognitive behaviors is the ability to maintain parametric information briefly in mind, in the time scale of seconds, to span delays between sensory information and actions. This component of working memory is fragile and quickly degrades with delay length. Under the assumption that behavioral delay-dependencies mark core functions of the working memory system, our goal is to find a neural circuit model that represents their neural mechanisms and apply it to research on working memory deficits in neuropsychiatric disorders. We have constrained computational models of spatial working memory with delay-dependent behavioral effects and with neural recordings in the prefrontal cortex during visuospatial working memory. I will show that a simple bump attractor model with weak inhomogeneities and short-term plasticity mechanisms can link neural data with fine-grained behavioral output in a trial-by-trial basis and account for the main delay-dependent limitations of working memory: precision, cardinal repulsion biases and serial dependence. I will finally present data from participants with neuropsychiatric disorders that suggest that serial dependence in working memory is specifically altered, and I will use the model to infer the possible neural mechanisms affected.

Discriminating distinct objects and concepts from sensory stimuli is essential for survival. Our brains accomplish this feat by forming meaningful internal representations in deep sensory networks with plastic synaptic connections. Experience-dependent plasticity presumably exploits temporal contingencies between sensory inputs to build these internal representations. However, the precise mechanisms underlying plasticity remain elusive. We derive a local synaptic plasticity model inspired by self-supervised machine learning techniques that shares a deep conceptual connection to Bienenstock-Cooper-Munro (BCM) theory and is consistent with experimentally observed plasticity rules. We show that our plasticity model yields disentangled object representations in deep neural networks without the need for supervision and implausible negative examples. In response to altered visual experience, our model qualitatively captures neuronal selectivity changes observed in the monkey inferotemporal cortex in-vivo. Our work suggests a plausible learning rule to drive learning in sensory networks while making concrete testable predictions.

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.

Neural representations of time are central to our understanding of the world around us. I review cognitive, neurophysiological and theoretical work that converges on three simple ideas. First, the time of past events is remembered via populations of neurons with a continuum of functional time constants. Second, these time constants evenly tile the log time axis. This results in a neural Weber-Fechner scale for time which can support behavioral Weber-Fechner laws and characteristic behavioral effects in memory experiments. Third, these populations appear as dual pairs---one type of population contains cells that change firing rate monotonically over time and a second type of population that has circumscribed temporal receptive fields. These ideas can be used to build artificial neural networks that have novel properties. Of particular interest, a convolutional neural network built using these principles can generalize to arbitrary rescaling of its inputs. That is, after learning to perform a classification task on a time series presented at one speed, it successfully classifies stimuli presented slowed down or sped up. This result illustrates the point that this confluence of ideas originating in cognitive psychology and measured in the mammalian brain could have wide-reaching impacts on AI research.

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.

The fusion of cell aggregates widely exists during biological processes such as development, tissue regeneration, and tumor invasion. Cellular spheroids (spherical cell aggregates) are commonly used to study this phenomenon. In previous studies, with approximated assumptions and measurements, researchers found that the fusion of two spheroids with some cell type is similar to the coalescence of two liquid droplets. However, with more accurate measurements focusing on the overall shape evolution in this process, we find that even in the previously-regarded liquid-like regime, the fusion process of spheroids can be very different from regular liquid coalescence. We conduct numerical simulations using both standard particulate models and vertex models with both Molecular Dynamics and Brownian Dynamics. The simulation results show that the difference between spheroids and regular liquid droplets is caused by the microscopic overdamped dynamics of each cell rather than the topological cell-cell interactions in the vertex model. Our research reveals the necessity of a new continuum theory for “liquid” with microscopically overdamped components, such as cellular and colloidal systems. Detailed analysis of our simulation results of different system sizes provides the basis for developing the new theory.

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.

We present new kinematic bending measures and quadratic energies for isotropic elastic plates and shells, with certain desirable features not present in commonly employed models in mechanics and soft matter. These are justified both by simple physical arguments related to the through-thickness variation in strain, and through a detailed reduction from a three-dimensional energy quadratic in stretch. The measure of plate bending is a dilation-invariant surface tensor that couples stretch and curvature in a natural extension of primitive generalized bending strains for straight rods. The extension to naturally-curved rods and shells, for which the pure stretching of a curved rest configuration is not a dilation, contrasts with previous ad hoc postulated forms. Our results provide a clean basis for simple models of low-dimensional elastic systems, and should enable more accurate probing of the structure of singularities in soft sheets and membranes.