Mathematical Modeling seminars

I will present my recent work on mathematical modeling of human memory. I will argue that memory recall of random lists of items is governed by the universal algorithm resulting in the analytical relation between the number of items in memory and the number of items that can be successfully recalled. The retention of items in memory on the other hand is not universal and differs for different types of items being remembered, in particular retention curves for words and sketches is different even when sketches are made to only carry information about an object being drawn. I will discuss the putative reasons for these observations and introduce the phenomenological model predicting retention curves.

Cognitive models of timing often include a pacemaker analogue whose ticks are accumulated to form an internal representation of time, and a threshold that determines when a target duration has elapsed. However, clear EEG manifestations of these abstract components have not yet been identified. We measured the EEG of subjects while they performed a temporal bisection task in which they were requested to categorize visual stimuli as short or long in duration. We report an ERP component whose amplitude depends monotonically on the stimulus duration. The relation of the ERP amplitude and stimulus duration can be captured by a simple model, adapted from a known drift-diffusion model for time perception. It includes a noisy accumulator that starts with the stimulus onset and a threshold. If the threshold is reached during stimulus presentation, the stimulus is categorized as "long", otherwise the stimulus is categorized as "short". At the stimulus offset, a response proportional to the distance to the threshold is emitted. This simple model has two parameters that fit both the behavior and ERP amplitudes recorded in the task. Two subsequent experiments replicate and extend this finding to another modality (touch) as well as to different time ranges (subsecond and suprasecond), establishing the described ERP component as a useful handle on the cognitive processes involved in temporal decisions.

The visual system is characterized by multiple mirrored visuotopic maps, with each repetition corresponding to a different visual area. In this work we explore whether such visuotopic organization can emerge as a result of minimizing the total wire length between neurons connected in a deep hierarchical network. Our results show that networks with purely feedforward connectivity typically result in a single visuotopic map, and in certain cases no visuotopic map emerges. However, when we modify the network by introducing lateral connections, with sufficient lateral connectivity among neurons within layers, multiple visuotopic maps emerge, where some connectivity motifs yield mirrored alternations of visuotopic maps–a signature of biological visual system areas. These results demonstrate that different connectivity profiles have different emergent organizations under the minimum total wire length hypothesis, and highlight that characterizing the large-scale spatial organizing of tuning properties in a biological system might also provide insights into the underlying connectivity.

Measuring the degree of consciousness an organism possesses has remained a longstanding challenge in Neuroscience. In part, this is due to the difficulty of finding the appropriate mathematical tools for describing such a subjective phenomenon. Current methods relate the level of consciousness to the complexity of neural activity, i.e., using the information contained in a stream of recorded signals they can tell whether the subject might be awake, asleep, or anaesthetised. Usually, the signals stemming from a complex system are correlated in time; the behaviour of the future depends on the patterns in the neural activity of the past. However these past-future relationships remain either hidden to, or not taken into account in the current measures of consciousness. These past-future correlations are likely to contain more information and thus can reveal a richer understanding about the behaviour of complex systems like a brain. Our work employs the "epsilon-machines” framework to account for the time correlations in neural recordings. In a nutshell, epsilon-machines reveal how much of the past neural activity is needed in order to accurately predict how the activity in the future will behave, and this is summarised in a single number called "statistical complexity". If a lot of past neural activity is required to predict the future behaviour, then can we say that the brain was more “awake" at the time of recording? Furthermore, if we read the recordings in reverse, does the difference between forward and reverse-time statistical complexity allow us to quantify the level of time asymmetry in the brain? Neuroscience predicts that there should be a degree of time asymmetry in the brain. However, this has never been measured. To test this, we used neural recordings measured from the brains of fruit flies and inferred the epsilon-machines. We found that the nature of the past and future correlations of neural activity in the brain, drastically changes depending on whether the fly was awake or anaesthetised. Not only does our study find that wakeful and anaesthetised fly brains are distinguished by how statistically complex they are, but that the amount of correlations in wakeful fly brains was much more sensitive to whether the neural recordings were read forward vs. backwards in time, compared to anaesthetised brains. In other words, wakeful fly brains were more complex, and time asymmetric than anaesthetised ones.

A major unresolved question in microbiome research is whether the complex ecological patterns observed in surveys of natural communities can be explained and predicted by fundamental, quantitative principles. Bridging theory and experiment is hampered by the multiplicity of ecological processes that simultaneously affect community assembly and a lack of theoretical tools for modeling diverse ecosystems. Here, I will present a simple ecological model of microbial communities that reproduces large-scale ecological patterns observed across multiple natural and experimental settings including compositional gradients, clustering by environment, diversity/harshness correlations, and nestedness. Surprisingly, our model works despite having a “random metabolisms” and “random consumer preferences”. This raises the natural of question of why random ecosystems can describe real-world experimental data. In the second, more theoretical part of the talk, I will answer this question by showing that when a community becomes diverse enough, it will always self-organize into a stable state whose properties are well captured by a “typical random ecosystems”.

When making decision under risk, people often exhibit behaviours that classical economic theories cannot explain. Newer models that attempt to account for these ‘irrational’ behaviours often lack neuroscience bases and require the introduction of subjective and problem-specific constructs. Here, we present a decision-making model inspired by the prediction error signals and introspective neuronal replay reported in the brain. In the model, decisions are chosen based on ‘anticipated surprise’, defined by a nonlinear average of the differences between individual outcomes and a reference point. The reference point is determined by the expected value of the possible outcomes, which can dynamically change during the mental simulation of decision-making problems involving sequential stages. Our model elucidates the contribution of each stage to the appeal of available options in a decision-making problem. This allows us to explain several economic paradoxes and gambling behaviours. Our work could help bridge the gap between decision-making theories in economics and neurosciences.

Recent advances of single cell techniques catalyzed quantitative studies on the dynamics of cell phenotypic transitions (CPT) emerging as a new field. However, fixed cell-based approaches have fundamental limits on revealing temporal information, and fluorescence-based live cell imaging approaches are technically challenging for multiplex long-term imaging. To tackle the challenges, we developed an integrated experimental/computational platform for reconstructing single cell phenotypic transition dynamics. Experimentally, we developed a live-cell imaging platform to record the phenotypic transition path of A549 VIM-RFP reporter cell line and unveil parallel paths of epithelial-to-mesenchymal transition (EMT). Computationally, we modified a finite temperature string method to reconstruct the reaction coordinate from the paths, and reconstruct a corresponding quasi-potential, which reveals that the EMT process resembles a barrier-less relaxation process. Our work demonstrates the necessity of extracting dynamical information of phenotypic transitions and the existence of a unified theoretical framework describing transition and relaxation dynamics in systems with and without detailed balance.

Inhibitory neurons take on many forms and functions. How this diversity contributes to memory function is not completely known. Previous formal studies indicate inhibition differentiated by local and global connectivity in associative memory networks functions to rescale the level of retrieval of excitatory assemblies. However, such studies lack biological details such as a distinction between types of neurons (excitatory and inhibitory), unrealistic connection schemas, and non-sparse assemblies. In this study, we present a rate-based cortical model where neurons are distinguished (as excitatory, local inhibitory, or global inhibitory), connected more realistically, and where memory items correspond to sparse excitatory assemblies. We use this model to study how local-global inhibition balance can alter memory retrieval in associative memory structures, including naturalistic and artificial structures. Experimental studies have reported inhibitory neurons and their sub-types uniquely respond to specific stimuli and can form sophisticated, joint excitatory-inhibitory assemblies. Our model suggests such joint assemblies, as well as a distribution and rebalancing of overall inhibition between two inhibitory sub-populations – one connected to excitatory assemblies locally and the other connected globally – can quadruple the range of retrieval across related memories. We identify a possible functional role for local-global inhibitory balance to, in the context of choice or preference of relationships, permit and maintain a broader range of memory items when local inhibition is dominant and conversely consolidate and strengthen a smaller range of memory items when global inhibition is dominant. This model therefore highlights a biologically-plausible and behaviourally-useful function of inhibitory diversity in memory.

Predictive coding represents a promising framework for understanding brain function, postulating that the brain continuously inhibits predictable sensory input, ensuring a preferential processing of surprising elements. A central aspect of this view on cortical computation is its hierarchical connectivity, involving recurrent message passing between excitatory bottom-up signals and inhibitory top-down feedback. Here we use computational modelling to demonstrate that such architectural hard-wiring is not necessary. Rather, predictive coding is shown to emerge as a consequence of energy efficiency, a fundamental requirement of neural processing. When training recurrent neural networks to minimise their energy consumption while operating in predictive environments, the networks self-organise into prediction and error units with appropriate inhibitory and excitatory interconnections and learn to inhibit predictable sensory input. We demonstrate that prediction units can reliably be identified through biases in their median preactivation, pointing towards a fundamental property of prediction units in the predictive coding framework. Moving beyond the view of purely top-down driven predictions, we demonstrate via virtual lesioning experiments that networks perform predictions on two timescales: fast lateral predictions among sensory units and slower prediction cycles that integrate evidence over time. Our results, which replicate across two separate data sets, suggest that predictive coding can be interpreted as a natural consequence of energy efficiency. More generally, they raise the question which other computational principles of brain function can be understood as a result of physical constraints posed by the brain, opening up a new area of bio-inspired, machine learning-powered neuroscience research.

Recently, the field of computational neuroscience has seen an explosion of the use of trained recurrent network models (RNNs) to model patterns of neural activity. These RNN models are typically characterized by tuned recurrent interactions between rate 'units' whose dynamics are governed by smooth, continuous differential equations. However, the response of biological single neurons is better described by all-or-none events - spikes - that are triggered in response to the processing of their synaptic input by the complex dynamics of their membrane. One line of research has attempted to resolve this discrepancy by linking the average firing probability of a population of simplified spiking neuron models to rate dynamics similar to those used for RNN units. However, challenges remain to account for complex temporal dependencies in the biological single neuron response and for the heterogeneity of synaptic input across the population. Here, we make progress by showing how to derive dynamic rate equations for a population of spiking neurons with multi-timescale adaptation properties - as this was shown to accurately model the response of biological neurons - while they receive independent time-varying inputs, leading to plausible asynchronous activity in the network. The resulting rate equations yield an insightful segregation of the population's response into dynamics that are driven by the mean signal received by the neural population, and dynamics driven by the variance of the input across neurons, with respective timescales that are in agreement with slice experiments. Further, these equations explain how input variability can shape log-normal instantaneous rate distributions across neurons, as observed in vivo. Our results help interpret properties of the neural population response and open the way to investigating whether the more biologically plausible and dynamically complex rate model we derive could provide useful inductive biases if used in an RNN to solve specific tasks.

Low-dimensional population dynamics can be observed in neural activity recorded from the prefrontal cortex (PFC) of subjects performing simple cognitive tasks. Many studies have shown that recurrent neural networks (RNNs) trained on the same tasks can reproduce qualitatively these state space trajectories, and have used them as models of how neuronal dynamics implement task computations. The PFC is also viewed as a conductor that organizes the communication between cortical areas and provides contextual information. It is then not clear what is its role in solving simple cognitive tasks. Do the low-dimensional trajectories observed in the PFC really correspond to the computations that it performs? Or do they indirectly reflect the computations occurring within the cortical areas projecting to the PFC? To address these questions, we modelled cortical areas with a modular RNN and equipped it with a PFC-like cognitive system. When trained on cognitive tasks, this multi-system brain model can reproduce the low-dimensional population responses observed in neuronal activity as well as classical RNNs. Qualitatively different mechanisms can emerge from the training process when varying some details of the architecture such as the time constants. In particular, there is one class of models where it is the dynamics of the cognitive system that is implementing the task computations, and another where the cognitive system is only necessary to provide contextual information about the task rule as task performance is not impaired when preventing the system from accessing the task inputs. These constitute two different hypotheses about the causal role of the PFC in solving simple cognitive tasks, which could motivate further experiments on the brain.

Spike Timing Dependent Plasticity (STDP) is argued to modulate synaptic strength depending on the timing of pre- and postsynaptic spikes. Physiological experiments identified a variety of temporal kernels: Hebbian, anti-Hebbian and symmetrical LTP/LTD. In this work we present a novel plasticity model, the Voltage-Dependent Pathway Model (VDP), which is able to replicate those distinct kernel types and intermediate versions with varying LTP/LTD ratios and symmetry features. In addition, unlike previous models it retains these characteristics for different neuron models, which allows for comparison of plasticity in different neuron types. The plastic updates depend on the relative strength and activation of separately modeled LTP and LTD pathways, which are modulated by glutamate release and postsynaptic voltage. We used the 15 neuron type parametrizations in the GLIF5 model presented by Teeter et al. (2018) in combination with the VDP to simulate a range of standard plasticity protocols including standard STDP experiments, frequency dependency experiments and low frequency stimulation protocols. Slight variation in kernel stability and frequency effects can be identified between the neuron types, suggesting that the neuron type may have an effect on the effective learning rule. This plasticity model builds a middle ground between biophysical and phenomenological models allowing not just for the combination with more complex and biophysical neuron models, but is also computationally efficient so can be used in network simulations. Therefore it offers the possibility to explore the functional role of the different kernel types and electrophysiological differences in heterogeneous networks in future work.

Decision-making in noisy environments with constant sensory evidence involves integrating sequentially-sampled evidence, a strategy formalized by diffusion models which is supported by decades behavioral and neural findings. By contrast, it is unknown whether this strategy is also used during decision-making when the underlying sensory evidence is expected to change. Here, we trained monkeys to identify the dominant color of a dynamically refreshed checkerboard pattern that doesn't become informative until after a variable delay. Animals' behavioral responses were briefly suppressed after an abrupt change in evidence, and many neurons in the frontal eye field displayed a corresponding dip in activity at this time, similar to the dip frequently observed after stimulus onset. Generalized drift-diffusion models revealed that behavior and neural activity were consistent with a brief suppression of motor output without a change in evidence accumulation itself, in contrast to the popular belief that evidence accumulation is paused or reset. These results suggest that a brief interruption in motor preparation is an important strategy for dealing with changing evidence during perceptual decision making.

It is common for animals to use self-generated movements to actively sense the surrounding environment. For instance, rodents rhythmically move their whiskers to explore the space close to their body. The mouse whisker system has become a standard model to study active sensing and sensorimotor integration through feedback loops. In this work, we developed a bioinspired spiking neural network model of the sensorimotor peripheral whisker system, modelling trigeminal ganglion, trigeminal nuclei, facial nuclei, and central pattern generator neuronal populations. This network was embedded in a virtual mouse robot, exploiting the Neurorobotics Platform, a simulation platform offering a virtual environment to develop and test robots driven by brain-inspired controllers. Eventually, the peripheral whisker system was properly connected to an adaptive cerebellar network controller. The whole system was able to drive active whisking with learning capability, matching neural correlates of behaviour experimentally recorded in mice.

Several lines of research have demonstrated that internally generated sensory experience - such as during memory, dreaming and mental imagery - activates similar neural representations as externally triggered perception. This overlap raises a fundamental challenge: how is the brain able to keep apart signals reflecting imagination and reality? In a series of online psychophysics experiments combined with computational modelling, we investigated to what extent imagination and perception are confused when the same content is simultaneously imagined and perceived. We found that simultaneous congruent mental imagery consistently led to an increase in perceptual presence responses, and that congruent perceptual presence responses were in turn associated with a more vivid imagery experience. Our findings can be best explained by a simple signal detection model in which imagined and perceived signals are added together. Perceptual reality monitoring can then easily be implemented by evaluating whether this intermixed signal is strong or vivid enough to pass a ‘reality threshold’. Our model suggests that, in contrast to self-generated sensory changes during movement, our brain does not discount self-generated sensory signals during mental imagery. This has profound implications for our understanding of reality monitoring and perception in general.

Recent advances in optogenetics allow for stimulation of neurons with sub-millisecond spike jitter and single neuron selectivity. Already this precision has revealed new levels of cortical sensitivity: stimulating tens of neurons can yield changes in the mean firing rate of thousands of similarly tuned neurons. This extreme sensitivity suggests that cortical dynamics are near criticality. Criticality is often studied in neural systems as a non-equilibrium thermodynamic process in which scale-free patterns of activity, called avalanches, emerge between distinct states of spontaneous activity. While criticality is well studied, it is still unclear what these distinct states of spontaneous activity are and what responses we expect from stimulation of this activity. By answering these questions, optogenetic stimulation will become a new avenue for approaching criticality and understanding cortical dynamics. Here, for the first time, we study the effects of optogenetic-like stimulation on a model near criticality. We study a model of Inhibitory/Excitatory (I/E) Leaky Integrate and Fire (LIF) spiking neurons which display a region of high sensitivity as seen in experiments. We find that this region of sensitivity is, indeed, near criticality. We derive the Dynamic Mean Field Theory of this model and find that the distinct states of activity are asynchrony and synchrony. We use our theory to characterize response to various types and strengths of optogenetic stimulation. Our model and theory predict that asynchronous, near-critical dynamics can have two qualitatively different responses to stimulation: one characterized by high sensitivity, discrete event responses, and high trial-to-trial variability, and another characterized by low sensitivity, continuous responses with characteristic frequencies, and low trial-to-trial variability. While both response types may be considered near-critical in model space, networks which are closest to criticality show a hybrid of these response effects.

The brain exhibits a rich repertoire of oscillatory patterns organized in space, time and frequency. However, despite ever more-detailed characterizations of spectrally-resolved network patterns, the principles governing oscillatory activity at the system-level remain unclear. Here, we propose that the transient emergence of spatially organized brain rhythms are signatures of weakly stable synchronization between subsets of brain areas, naturally occurring at reduced collective frequencies due to the presence of time delays. To test this mechanism, we build a reduced network model representing interactions between local neuronal populations (with damped oscillatory response at 40Hz) coupled in the human neuroanatomical network. Following theoretical predictions, weakly stable cluster synchronization drives a rich repertoire of short-lived (or metastable) oscillatory modes, whose frequency inversely depends on the number of units, the strength of coupling and the propagation times. Despite the significant degree of reduction, we find a range of model parameters where the frequencies of collective oscillations fall in the range of typical brain rhythms, leading to an optimal fit of the power spectra of magnetoencephalographic signals from 89 heathy individuals. These findings provide a mechanistic scenario for the spontaneous emergence of frequency-specific long-range phase-coupling observed in magneto- and electroencephalographic signals as signatures of resonant modes emerging in the space-time structure of the Connectome, reinforcing the importance of incorporating realistic time delays in network models of oscillatory brain activity.

A myriad of pathological changes associated with epilepsy, including the loss of specific cell types, improper expression of individual ion channels, and synaptic sprouting, can be recast as decreases in cell and circuit heterogeneity. In recent experimental work, we demonstrated that biophysical diversity is a key characteristic of human cortical pyramidal cells, and past theoretical work has shown that neuronal heterogeneity improves a neural circuit’s ability to encode information. Viewed alongside the fact that seizure is an information-poor brain state, these findings motivate the hypothesis that epileptogenesis can be recontextualized as a process where reduction in cellular heterogeneity renders neural circuits less resilient to seizure onset. By comparing whole-cell patch clamp recordings from layer 5 (L5) human cortical pyramidal neurons from epileptogenic and non-epileptogenic tissue, we present the first direct experimental evidence that a significant reduction in neural heterogeneity accompanies epilepsy. We directly implement experimentally-obtained heterogeneity levels in cortical excitatory-inhibitory (E-I) stochastic spiking network models. Low heterogeneity networks display unique dynamics typified by a sudden transition into a hyper-active and synchronous state paralleling ictogenesis. Mean-field analysis reveals a distinct mathematical structure in these networks distinguished by multi-stability. Furthermore, the mathematically characterized linearizing effect of heterogeneity on input-output response functions explains the counter-intuitive experimentally observed reduction in single-cell excitability in epileptogenic neurons. This joint experimental, computational, and mathematical study showcases that decreased neuronal heterogeneity exists in epileptogenic human cortical tissue, that this difference yields dynamical changes in neural networks paralleling ictogenesis, and that there is a fundamental explanation for these dynamics based in mathematically characterized effects of heterogeneity. These interdisciplinary results provide convincing evidence that biophysical diversity imbues neural circuits with resilience to seizure and a new lens through which to view epilepsy, the most common serious neurological disorder in the world, that could reveal new targets for clinical treatment.