Mathematical Modeling seminars

Optimally interacting with a changeable and uncertain world requires estimating and representing uncertainty. Psychiatric and neurodevelopmental conditions such as anxiety and autism are characterized by an altered response to uncertainty. I will review the evidence for these phenomena from computational modelling, and outline the planned experiments from our lab to add further weight to these ideas. If time allows, I will present results from a control sample in a novel task interrogating a particular type of uncertainty and their associated transdiagnostic psychiatric traits.

The typical goal of homeostatic mechanisms is to ensure a system operates at or in the vicinity of a stable set point, where a particular measure is relatively constant and stable. Neural firing rate homeostasis is unusual in that a set point of fixed firing rate is at odds with the goal of a neuron to convey information, or produce timed motor responses, which require temporal variations in firing rate. Therefore, for a neuron, a range of firing rates is required for optimal function, which could, for example, be set by a dual system that controls both mean and variance of firing rate. We explore, both via simulations and analysis, how two experimentally measured mechanisms for firing rate homeostasis can cooperate to improve information processing and avoid the pitfall of pulling in different directions when their set points do not appear to match.

The human brain is a complex network of neuronal connections. The precise arrangement of these connections, otherwise known as the topology of the network, is crucial to its functioning. Recent efforts to understand how the complex topology of the brain has emerged have used generative mathematical models, which grow synthetic networks according to specific wiring rules. Evidence suggests that a wiring rule which emulates a trade-off between connection costs and functional benefits can produce networks that capture essential topological properties of brain networks. In this webinar, Professor Alex Fornito and Dr Stuart Oldham will discuss these previous findings, as well as their own efforts in creating more physiologically constrained generative models. Professor Alex Fornito is Head of the Brain Mapping and Modelling Research Program at the Turner Institute for Brain and Mental Health. His research focuses on developing new imaging techniques for mapping human brain connectivity and applying these methods to shed light on brain function in health and disease. Dr Stuart Oldham is a Research Fellow at the Turner Institute for Brain and Mental Health and a Research Officer at the Murdoch Children’s Research Institute. He is interested in characterising the organisation of human brain networks, with particular focus on how this organisation develops, using neuroimaging and computational tools.

The human brain is a complex organ characterized by heterogeneous patterns of interconnections. Non-invasive imaging techniques now allow for these patterns to be carefully and comprehensively mapped in individual humans, paving the way for a better understanding of how wiring supports cognitive processes. While a large body of work now focuses on descriptive statistics to characterize these wiring patterns, a critical open question lies in how the organization of these networks constrains the potential repertoire of brain dynamics. In this talk, I will describe an approach for understanding how perturbations to brain dynamics propagate through complex wiring patterns, driving the brain into new states of activity. Drawing on a range of disciplinary tools – from graph theory to network control theory and optimization – I will identify control points in brain networks and characterize trajectories of brain activity states following perturbation to those points. Finally, I will describe how these computational tools and approaches can be used to better understand the brain's intrinsic control mechanisms and their alterations in psychiatric conditions.

There is little consensus on the level of spatial complexity at which dendrites operate. On the one hand, emergent evidence indicates that synapses cluster at micrometer spatial scales. On the other hand, most modelling and network studies ignore dendrites altogether. This dichotomy raises an urgent question: what is the smallest relevant spatial scale for understanding dendritic computation? We have developed a method to construct compartmental models at any level of spatial complexity. Through carefully chosen parameter fits, solvable in the least-squares sense, we obtain accurate reduced compartmental models. Thus, we are able to systematically construct passive as well as active dendrite models at varying degrees of spatial complexity. We evaluate which elements of the dendritic computational repertoire are captured by these models. We show that many canonical elements of the dendritic computational repertoire can be reproduced with few compartments. For instance, for a model to behave as a two-layer network, it is sufficient to fit a reduced model at the soma and at locations at the dendritic tips. In the basal dendrites of an L2/3 pyramidal model, we reproduce the backpropagation of somatic action potentials (APs) with a single dendritic compartment at the tip. Further, we obtain the well-known Ca-spike coincidence detection mechanism in L5 Pyramidal cells with as few as eleven compartments, the requirement being that their spacing along the apical trunk supports AP backpropagation. We also investigate whether afferent spatial connectivity motifs admit simplification by ablating targeted branches and grouping affected synapses onto the next proximal dendrite. We find that voltage in the remaining branches is reproduced if temporal conductance fluctuations stay below a limit that depends on the average difference in input resistance between the ablated branches and the next proximal dendrite. Consequently, when the average conductance load on distal synapses is constant, the dendritic tree can be simplified while appropriately decreasing synaptic weights. When the conductance level fluctuates strongly, for instance through a-priori unpredictable fluctuations in NMDA activation, a constant weight rescale factor cannot be found, and the dendrite cannot be simplified. We have created an open source Python toolbox (NEAT - https://neatdend.readthedocs.io/en/latest/) that automatises the simplification process. A NEST implementation of the reduced models, currently under construction, will enable the simulation of few-compartment models in large-scale networks, thus bridging the gap between cellular and network level neuroscience.

The ability to simultaneously record the activity from tens to thousands to tens of thousands of neurons has allowed us to analyze the computational role of population activity as opposed to single neuron activity. Recent work on a variety of cortical areas suggests that neural function may be built on the activation of population-wide activity patterns, the neural modes, rather than on the independent modulation of individual neural activity. These neural modes, the dominant covariation patterns within the neural population, define a low dimensional neural manifold that captures most of the variance in the recorded neural activity. We refer to the time-dependent activation of the neural modes as their latent dynamics. As an example, we focus on the ability to execute learned actions in a reliable and stable manner. We hypothesize that the ability to perform a given behavior in a consistent manner requires that the latent dynamics underlying the behavior also be stable. The stable latent dynamics, once identified, allows for the prediction of various behavioral features, using models whose parameters remain fixed throughout long timespans. We posit that latent cortical dynamics within the manifold are the fundamental and stable building blocks underlying consistent behavioral execution.

To understand how the brain can perform complex tasks such as perception, we have to understand how information enters the brain, how it is transformed and how it is transferred. But, how do we measure information transfer in the brain? This presentation will start with a general introduction of what mutual information is and how to measure it in an experimental setup. Next, the talk will focus on how this can be used to develop brain models at different (spatial) levels, from the microscopic single neuron level to the macroscopic network and behavioural level. How can we incorporate the knowledge about single neurons, that already show complex dynamics, into network activity and link this to behaviour?

In the past several years, I have been involved in building a biologically realistic model of the monkey visual cortex. Work on one of the input layers (4Ca) of the primary visual cortex (V1) is now nearly complete, and I would like to share some of what I have learned with the community. After a brief overview of the model and its capabilities, I would like to focus on three sets of results that represent three different aspects of the modeling. They are: (i) emergent E-I dynamics in local circuits; (ii) how visual cortical neurons acquire their ability to detect edges and directions of motion, and (iii) a view across the cortical surface: nonequilibrium steady states (in analogy with statistical mechanics) and beyond.

The assumed data generating models (response distributions) of experimental or observational data in cognitive science have become increasingly complex over the past decades. This trend follows a revolution in model estimation methods and a drastic increase in computing power available to researchers. Today, higher-level cognitive functions can well be captured by and understood through computational cognitive models, a common example being drift diffusion models for decision processes. Such models are often expressed as the combination of two modeling layers. The first layer is the response distribution with corresponding distributional parameters tailored to the cognitive process under investigation. The second layer are latent models of the distributional parameters that capture how those parameters vary as a function of design, stimulus, or person characteristics, often in an additive manner. Such cognitive models can thus be understood as special cases of distributional regression models where multiple distributional parameters, rather than just a single centrality parameter, are predicted by additive models. Because of their complexity, distributional models are quite complicated to estimate, but recent advances in Bayesian estimation methods and corresponding software make them increasingly more feasible. In this talk, I will speak about the specification, estimation, and post-processing of Bayesian distributional regression models and how they can help to better understand cognitive processes.

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no cure. Yet, a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages associated with various cognitive deficits and pathologies. How can we use mathematical modelling to gain insight into this process and, doing so, gain understanding about how the brain works? In this talk, I will show that by linking new mathematical theories to recent progress in imaging, we can unravel some of the universal features associated with dementia and, more generally, brain functions.

The Stabilized Supralinear Network is a model of recurrently connected excitatory (E) and inhibitory (I) neurons with a supralinear input-output relation. It can explain cortical computations such as response normalization and inhibitory stabilization. However, the network's connectivity is designed by hand, based on experimental measurements. How the recurrent synaptic weights can be learned from the sensory input statistics in a biologically plausible way is unknown. Earlier theoretical work on plasticity focused on single neurons and the balance of excitation and inhibition but did not consider the simultaneous plasticity of recurrent synapses and the formation of receptive fields. Here we present a recurrent E-I network model where all synaptic connections are simultaneously plastic, and E neurons self-stabilize by recruiting co-tuned inhibition. Motivated by experimental results, we employ a local Hebbian plasticity rule with multiplicative normalization for E and I synapses. We develop a theoretical framework that explains how plasticity enables inhibition balanced excitatory receptive fields that match experimental results. We show analytically that sufficiently strong inhibition allows neurons' receptive fields to decorrelate and distribute themselves across the stimulus space. For strong recurrent excitation, the network becomes stabilized by inhibition, which prevents unconstrained self-excitation. In this regime, external inputs integrate sublinearly. As in the Stabilized Supralinear Network, this results in response normalization and winner-takes-all dynamics: when two competing stimuli are presented, the network response is dominated by the stronger stimulus while the weaker stimulus is suppressed. In summary, we present a biologically plausible theoretical framework to model plasticity in fully plastic recurrent E-I networks. While the connectivity is derived from the sensory input statistics, the circuit performs meaningful computations. Our work provides a mathematical framework of plasticity in recurrent networks, which has previously only been studied numerically and can serve as the basis for a new generation of brain-inspired unsupervised machine learning algorithms.

Odor memories are exceptionally robust and essential for the survival of many species. In rodents, the olfactory cortex shows features of an autoassociative memory network and plays a key role in the retrieval of olfactory memories (Meissner-Bernard et al., 2019). Interestingly, the telencephalic area Dp, the zebrafish homolog of olfactory cortex, transiently enters a state of precise balance during the presentation of an odor (Rupprecht and Friedrich, 2018). This state is characterized by large synaptic conductances (relative to the resting conductance) and by co-tuning of excitation and inhibition in odor space and in time at the level of individual neurons. Our aim is to understand how this precise synaptic balance affects memory function. For this purpose, we build a simplified, yet biologically plausible spiking neural network model of Dp using experimental observations as constraints: besides precise balance, key features of Dp dynamics include low firing rates, odor-specific population activity and a dominance of recurrent inputs from Dp neurons relative to afferent inputs from neurons in the olfactory bulb. To achieve co-tuning of excitation and inhibition, we introduce structured connectivity by increasing connection probabilities and/or strength among ensembles of excitatory and inhibitory neurons. These ensembles are therefore structural memories of activity patterns representing specific odors. They form functional inhibitory-stabilized subnetworks, as identified by the “paradoxical effect” signature (Tsodyks et al., 1997): inhibition of inhibitory “memory” neurons leads to an increase of their activity. We investigate the benefits of co-tuning for olfactory and memory processing, by comparing inhibitory-stabilized networks with and without co-tuning. We find that co-tuned excitation and inhibition improves robustness to noise, pattern completion and pattern separation. In other words, retrieval of stored information from partial or degraded sensory inputs is enhanced, which is relevant in light of the instability of the olfactory environment. Furthermore, in co-tuned networks, odor-evoked activation of stored patterns does not persist after removal of the stimulus and may therefore subserve fast pattern classification. These findings provide valuable insights into the computations performed by the olfactory cortex, and into general effects of balanced state dynamics in associative memory networks.

Neuronal variability is a reflection of recurrent circuitry and cellular physiology. The modulation of neuronal variability is a reliable signature of cognitive and processing state. A pervasive yet puzzling feature of cortical circuits is that despite their complex wiring, population-wide shared spiking variability is low dimensional with all neurons fluctuating en masse. We show that the spatiotemporal dynamics in a spatially structured network produce large population-wide shared variability. When the spatial and temporal scales of inhibitory coupling match known physiology, model spiking neurons naturally generate low dimensional shared variability that captures in vivo population recordings along the visual pathway. Further, we show that firing rate models with spatial coupling can also generate chaotic and low-dimensional rate dynamics. The chaotic parameter region expands when the network is driven by correlated noisy inputs, while being insensitive to the intensity of independent noise.

In vivo, the human genome folds into a characteristic ensemble of 3D structures. The mechanism driving the folding process remains unknown. A theoretical model for chromatin (the minimal chromatin model) explains the folding of interphase chromosomes and generates chromosome conformations consistent with experimental data is presented. The energy landscape of the model was derived by using the maximum entropy principle and relies on two experimentally derived inputs: a classification of loci into chromatin types and a catalog of the positions of chromatin loops. This model was generalized by utilizing a neural network to infer these chromatin types using epigenetic marks present at a locus, as assayed by ChIP-Seq. The ensemble of structures resulting from these simulations completely agree with HI-C data and exhibits unknotted chromosomes, phase separation of chromatin types, and a tendency for open chromatin to lie at the periphery of chromosome territories. Although this theoretical methodology was trained in one cell line, the human GM12878 lymphoblastoid cells, it has successfully predicted the structural ensembles of multiple human cell lines. Finally, going beyond Hi-C, our predicted structures are also consistent with microscopy measurements. Analysis of both structures from simulation and microscopy reveals that short segments of chromatin make two-state transitions between closed conformations and open dumbbell conformations. For gene active segments, the vast majority of genes appear clustered in the linker region of the chromatin segment, allowing us to speculate possible mechanisms by which chromatin structure and dynamics may be involved in controlling gene expression. * Supported by the NSF

The human brain network architecture can reveal crucial aspects of brain function and dysfunction. The topology of this network (known as the connectome) is shaped by a trade-off between wiring cost and network efficiency, and it has highly connected hub regions playing a prominent role in many brain disorders. By studying a landscape of plausible brain networks that preserve the wiring cost, fragile and resilient hubs can be identified. In this webinar, Dr Leonardo Gollo and Dr James Pang from Monash University will discuss this approach across the lifespan and some of its implications for neurodevelopmental and neurodegenerative diseases. Dr Leonardo Gollo is a Senior Research Fellow at the Turner Institute for Brain and Mental Health, School of Psychological Sciences, Monash University. He holds an ARC Future Fellowship and his research interests include brain modelling, systems neuroscience, and connectomics. Dr James Pang is a Research Fellow at the Turner Institute for Brain and Mental Health, School of Psychological Sciences, Monash University. His research interests are on combining neuroimaging and biophysical modelling to better understand the mechanisms of brain function in health and disease.

Start of the talk will be 17:15h (CEST). This session is a double feature of the Cologne Theoretical Neuroscience Forum and the BCCN Berlin.

Many microorganisms and cells function in complex (non-Newtonian) fluids, which are mixtures of different materials and exhibit both viscous and elastic stresses. For example, mammalian sperm swim through cervical mucus on their journey through the female reproductive tract, and they must penetrate the viscoelastic gel outside the ovum to fertilize. In micro-scale swimming the dynamics emerge from the coupled interactions between the complex rheology of the surrounding media and the passive and active body dynamics of the swimmer. We use computational models of swimmers in viscoelastic fluids to investigate and provide mechanistic explanations for emergent swimming behaviors. I will discuss how flexible filaments (such as flagella) can store energy from a viscoelastic fluid to gain stroke boosts due to fluid elasticity. I will also describe 3D simulations of model organisms such as C. Reinhardtii and mammalian sperm, where we use experimentally measured stroke data to separate naturally coupled stroke and fluid effects. We explore why strokes that are adapted to Newtonian fluid environments might not do well in viscoelastic environments.

Behavioural measurements are often overlooked by computational neuroscientists, who prefer to focus on electrophysiological recordings or neuroimaging data. This attitude is largely due to perceived lack of depth/richness in relation to behavioural datasets. I will show how contemporary psychophysics can deliver extremely rich and highly constraining datasets that naturally interface with computational modelling. More specifically, I will demonstrate how psychophysics can be used to guide/constrain/refine computational models, and how models can be exploited to design/motivate/interpret psychophysical experiments. Examples will span a wide range of topics (from feature detection to natural scene understanding) and methodologies (from cascade models to deep learning architectures).

Though the term has many definitions, computational neuroscience is mainly about applying mathematics to the study of the brain. The brain—a jumble of all different kinds of neurons interconnected in countless ways that somehow produce consciousness—has been described as “the most complex object in the known universe”. Physicists for centuries have turned to mathematics to properly explain some of the most seemingly simple processes in the universe—how objects fall, how water flows, how the planets move. Equations have proved crucial in these endeavors because they capture relationships and make precise predictions possible. How could we expect to understand the most complex object in the universe without turning to mathematics? — The answer is we can’t, and that is why I wrote this book. While I’ve been studying and working in the field for over a decade, most people I encounter have no idea what “computational neuroscience” is or that it even exists. Yet a desire to understand how the brain works is a common and very human interest. I wrote this book to let people in on the ways in which the brain will ultimately be understood: through mathematical and computational theories. — At the same time, I know that both mathematics and brain science are on their own intimidating topics to the average reader and may seem downright prohibitory when put together. That is why I’ve avoided (many) equations in the book and focused instead on the driving reasons why scientists have turned to mathematical modeling, what these models have taught us about the brain, and how some surprising interactions between biologists, physicists, mathematicians, and engineers over centuries have laid the groundwork for the future of neuroscience. — Each chapter of Models of the Mind covers a separate topic in neuroscience, starting from individual neurons themselves and building up to the different populations of neurons and brain regions that support memory, vision, movement and more. These chapters document the history of how mathematics has woven its way into biology and the exciting advances this collaboration has in store.

The ability to simultaneously record the activity from tens to thousands and maybe even tens of thousands of neurons has allowed us to analyze the computational role of population activity as opposed to single neuron activity. Recent work on a variety of cortical areas suggests that neural function may be built on the activation of population-wide activity patterns, the neural modes, rather than on the independent modulation of individual neural activity. These neural modes, the dominant covariation patterns within the neural population, define a low dimensional neural manifold that captures most of the variance in the recorded neural activity. We refer to the time-dependent activation of the neural modes as their latent dynamics, and argue that latent cortical dynamics within the manifold are the fundamental and stable building blocks of neural population activity.