Mathematical Modeling seminars

The neurobiological mechanisms of arousal and anesthesia remain poorly understood. Recent evidence highlights the key role of interactions between the cerebral cortex and the diffusely projecting matrix thalamic nuclei. Here, we interrogate these processes in a whole-brain corticothalamic neural mass model endowed with targeted and diffusely projecting thalamocortical nuclei inferred from empirical data. This model captures key features seen in propofol anesthesia, including diminished network integration, lowered state diversity, impaired susceptibility to perturbation, and decreased corticocortical coherence. Collectively, these signatures reflect a suppression of information transfer across the cerebral cortex. We recover these signatures of conscious arousal by selectively stimulating the matrix thalamus, recapitulating empirical results in macaque, as well as wake-like information processing states that reflect the thalamic modulation of largescale cortical attractor dynamics. Our results highlight the role of matrix thalamocortical projections in shaping many features of complex cortical dynamics to facilitate the unique communication states supporting conscious awareness.

Activity dependent synaptic plasticity is considered to be a primary mechanism underlying learning and memory. Yet it is unclear whether plasticity rules such as STDP measured in vitro apply in vivo. Network models with STDP predict that activity patterns (e.g., place-cell spatial selectivity) should change much faster than observed experimentally. We address this gap by investigating a nonlinear calcium-based plasticity rule fit to experiments done in physiological conditions. In this model, LTP and LTD result from intracellular calcium transients arising almost exclusively from synchronous coactivation of pre- and postsynaptic neurons. We analytically approximate the full distribution of nonlinear calcium transients as a function of pre- and postsynaptic firing rates, and temporal correlations. This analysis directly relates activity statistics that can be measured in vivo to the changes in synaptic efficacy they cause. Our results highlight that both high-firing rates and temporal correlations can lead to significant changes to synaptic efficacy. Using a mean-field theory, we show that the nonlinear plasticity rule, without any fine-tuning, gives a stable, unimodal synaptic weight distribution characterized by many strong synapses which remain stable over long periods of time, consistent with electrophysiological and behavioral studies. Moreover, our theory explains how memories encoded by strong synapses can be preferentially stabilized by the plasticity rule. We confirmed our analytical results in a spiking recurrent network. Interestingly, although most synapses are weak and undergo rapid turnover, the fraction of strong synapses are sufficient for supporting realistic spiking dynamics and serve to maintain the network’s cluster structure. Our results provide a mechanistic understanding of how stable memories may emerge on the behavioral level from an STDP rule measured in physiological conditions. Furthermore, the plasticity rule we investigate is mathematically equivalent to other learning rules which rely on the statistics of coincidences, so we expect that our formalism will be useful to study other learning processes beyond the calcium-based plasticity rule.

One of Tali's cherished goals was to transform biology into physics. In his view, biologists were far too enamored by the details of the specific models they studied, losing sight of the big principles that may govern the behavior of these models. One such big principle that he suggested was the 'information bottleneck (IB) principle'. The iIB principle is an information-theoretical approach for extracting the relevant information that one random variable carries about another. Tali applied the IB principle to numerous problems in biology, gaining important insights in the process. Here I will describe two applications of the IB principle to neurobiological data. The first is the formalization of the notion of surprise that allowed us to rigorously estimate the memory duration and content of neuronal responses in auditory cortex, and the second is an application to behavior, allowing us to estimate 'optimal policies under information constraints' that shed interesting light on rat behavior.

CA3-CA1 synapses in the hippocampus are the initial locus of episodic memory. The action of acetylcholine alters cellular excitability, modifies neuronal networks, and triggers secondary signaling that directly affects long-term plasticity (LTP) (the cellular underpinning of memory). It is therefore considered a critical regulator of learning and memory in the brain. Its action via M4 metabotropic receptors in the presynaptic terminal of the CA3 neurons and M1 metabotropic receptors in the postsynaptic spines of CA1 neurons produce rich dynamics across multiple timescales. We developed a model to describe the activation of postsynaptic M1 receptors that leads to IP3 production from membrane PIP2 molecules. The binding of IP3 to IP3 receptors in the endoplasmic reticulum (ER) ultimately causes calcium release. This calcium release from the ER activates potassium channels like the calcium-activated SK channels and alters different aspects of synaptic signaling. In an independent signaling cascade, M1 receptors also directly suppress SK channels and the voltage-activated KCNQ2/3 channels, enhancing post-synaptic excitability. In the CA3 presynaptic terminal, we model the reduction of the voltage sensitivity of voltage-gated calcium channels (VGCCs) and the resulting suppression of neurotransmitter release by the action of the M4 receptors. Our results show that the reduced initial release probability because of acetylcholine alters short-term plasticity (STP) dynamics. We characterize the dichotomy of suppressing neurotransmitter release from CA3 neurons and the enhanced excitability of the postsynaptic CA1 spine. Mechanisms underlying STP operate over a few seconds, while those responsible for LTP last for hours, and both forms of plasticity have been linked with very distinct functions in the brain. We show that the concurrent suppression of neurotransmitter release and increased sensitivity conserves neurotransmitter vesicles and enhances the reliability in plasticity. Our work establishes a relationship between STP and LTP coordinated by neuromodulation with acetylcholine.

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.

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.