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Milnerwood, A. J. & Raymond, L. A. Early synaptic pathophysiology in neurodegeneration: Insights from Huntington’s disease. Trends Neurosci. 33(11), 513–523. https://doi.org/10.1016/j.tins.2010.08.002 (2010). In the last decade, increasing attention has been devoted to elucidating the connectivity matrices of neuronal circuits (connectomics). The development of advanced imaging methods has allowed this issue to be approached experimentally 68, but a detailed description of the architecture of extended circuits is not yet possible. With the method proposed here, the resulting in-degree and out-degree distributions are consistent with those expected from theoretical and experimental analysis 20, 57. Notably, this result has been obtained without any constraints on the degree distributions, which would be required for the generation of connections through randomized processes.

de Física, Facultad de Ciencias Exactas y Naturales, Instituto de Física de Buenos Aires (IFIBA), CONICET, Universidad de Buenos Aires, Buenos Aires, Argentina Peters, A. & Feldman, M. The projection of the lateral geniculate nucleus to area 17 of the rat cerebral cortex I general description. J. Neurocytol. 5, 63–84 (1976). Zooming out from the intra-neuron synaptic level, inter-neuron communication emerges as a principal determinant of the dynamics. Information transmission is mainly based on the emission of action potentials. The mechanism of this flow of ions was first explained by the influential Hodgkin-Huxley equations and corresponding circuits. The electrical current of the equivalent circuit is described by four differential equations that incorporate membrane capacity and the gating variables of the channels ( Hodgkin and Huxley, 1952). While the Hodgkin-Huxley model agrees with a wide range of experiments ( Patlak and Ortiz, 1985; Traub et al., 1991) and continues to be a reference for models of ion channels, it needs to be simplified to be expandable to the models of the neuronal population. The main difficulty with the Hodgkin-Huxley model is that it requires solving a system of differential equations for each of the gating parameters of each of the single ion channels of a cell while there are more than 300 types of ion channels discovered as of today ( Gabashvili et al., 2007). Various relaxing assumptions have been proposed, one of which is to dismiss the time dependence of membrane conductance and the dynamics of the action potential by simply assuming the firing happens when the electrical input accumulated at the membrane exceeds a threshold ( Abbott and Kepler, 1990). The latter model is known as integrate-and-fire ( Stein and Hodgkin, 1967), and it comes in different flavors depending on the form of nonlinearity assumed for the dynamics of leaky or refractory synapses ( Michaels et al., 2016). These neurons included Ivy and Bistratified cells whose cell bodies are mainly located in the SR and diffusely project an axonal cloud from SLM to SO isotropically 39. Similarly, dendrites are preferentially oriented in the direction going from the SO to the SLM and tend to be confined inside the axonal cloud. In our model, axons and dendrites where both designed as single ellipsoids (Fig. 5).

Soltesz, I. & Losonczy, A. CA1 pyramidal cell diversity enabling parallel information processing in the hippocampus. Nat Neurosci. 21(4), 484–493. https://doi.org/10.1038/s41593-018-0118-0 (2018). In this section, we review brain models across different scales that are faithful to biological constraints. We focus primarily on the first column from the left in Figure 3, starting from the realistic models with mesoscopic details to more coarse-grained frameworks. 1.1. Modeling at the Synaptic Level

Sloviter, R. S. & Lømo, T. Updating the lamellar hypothesis of hippocampal organization. Fr. Neural Circuits 6, 102. https://doi.org/10.3389/fncir.2012.00102 (2012).A potential solution for narrowing this computation gap can be sought at the hardware level. An instance of such a dedicated pipeline is neuromorphic processing units (NPU) that are power efficient and take time and dynamics into the equation from the beginning. An NPU is an array of neurosynaptic cores that contain computing models (neurons) and memory within the same unit. In short, the advantage of using NPUs is that they resemble the brain more realistically than a CPU or GPU because of asynchronous (event-based) communication, extreme parallelism (100–1,000,000 cores), and low power consumption ( Eli, 2022). Their efficiency and robustness also result from the Physical proximity of the computing unit and memory. Below popular examples of such NPUs are listed. Each of them stemmed from different initiatives. In addition to the implicit assumption of the adequacy of training data, the explicit assumption that these models rely on is that the solution is parsimonious, i.e., there are few descriptive parameters. Despite some possibility of error with this assumption in given problems ( Su et al., 2017), it is particularly useful in having arbitrarily less complicated descriptions that are generalizable, interpretable, and less prone to overfitting. In this paper, we demonstrate why focusing on the multi-scale dynamics of the brain is essential for biologically plausible and explainable results. For this goal, we review a large spectrum of computational models for reconstructing neural dynamics developed by diverse scientific fields, such as biological neuroscience (biological models), physics, and applied mathematics (phenomenological models), as well as statistics and computer science (data-driven models). On this path, it is crucial to consider the uniqueness of neural dynamics and the shortcomings of data collection. Neural dynamics are different from other forms of physical time series. In general, neural ensembles diverge from many canonical examples of dynamical systems in the following ways: Neural Dynamics Is Different Total reliance on data is especially questionable when the data has significant complications (as discussed in the introduction). Opting for a methodology guided by patterns rather than prior knowledge is problematic in particular when the principle patterns of data arise from uninteresting phenomena such as the particular way a given facility may print out the brain images ( Ng, 2021). A relatively new class of dynamical frameworks combines differential equations with machine learning in a more explicit fashion. It is noteworthy to mention that by neural ODE here, we are referring to the term used in Chen et al. (2018). Neural ODEs are a family of deep neural networks that learn the governing differential equations of the system, not to be confused with the differential equations of neuronal dynamics. This class of frameworks has been used to model the dynamics of time-varying signals. They begin by assuming that the underlying dynamics follow a differential equation. They can then be used to discover the parameters of that differential equation by using standard optimization of deep neural networks. As is evident, such formulations are quite useful in modeling and analyzing brain dynamics, especially using deep networks. Below we describe some of the relevant works in this subfield. 3.2.2.1. Neural Ordinary Differential Equations

Key Contributions: The objective is to bridge a gap in the literature of computational neuroscience, dynamical systems, and AI and to review the usability of the proposed generative models concerning the limitation of data, the objective of the study and the problem definition, prior knowledge of the system, and sets of assumptions (see Figure 2). 1. Biophysical Models of Psychiatry, CHU Sainte-Justine Research Center, Mila-Quebec AI Institute, Université de Montréal, Montréal, QC, Canada

Introduction

Human Brain Project (HBP): aimed to realistically simulate the human brain in supercomputers ( Miller, 2011). Giacopelli, G., Migliore, M. & Tegolo, D. Graph-theoretical derivation of brain structural connectivity. Appl. Math. Comput. 377, 125520 (2020). Markram, H. et al. Reconstruction and simulation of neocortical microcircuitry. Cell 163(2), 456–492. https://doi.org/10.1016/j.cell.2015.09.029 (2015).

Large-scale neuronal network activity can be simulated by simplifying the description of the electrophysiological properties of the individual neurons. With this aim, point-like neurons, usually modelled as integrate-and-fire 40 or Izhikevich models 41, can be adopted to significantly reduce the computational effort 34. Importantly, the connectivity between cells must be preserved to allow the emergence of the correct functional organization of neuronal activity which in turn requires a specific connectivity rule. In our approach, the rules for the generation of connections between any two neurons were implemented assuming that every neuronal class is characterized by an average shape. Morphological analysis has been performed by collecting the experimentally reconstructed morphologies of CA1 neuron subtypes from the literature and from public databases such as neuromorpho ( http://neuromorpho.org/) 42, Allen Brain Institute 43 ( https://portal.brain-map.org), and Janelia Research Campus 44 ( http://mouselight.janelia.org/). In this analysis, we have assumed that each cell class could be represented as a combination of geometrical shapes (ellipsoids and cones). The parameterization of these shapes (axonal and dendritic extensions) was generated by creating normal distributions for each of the parameters with peaks corresponding to the average values derived from the analysis and half-widths of 10% of the peaks (Table 1-SM). The dimensions of axons and cones for each neuron were then randomly sampled through an automatic procedure within the parameter distributions. The modeling of neuronal morphologies as combinations of ellipsoids and cones mimics the cross-section volume of pre-synaptic axons and post-synaptic dendrites (See Fig. SM-1). The description of the geometrical shapes adopted for each neuronal class is detailed in the " Results" section. Price, C. J. et al. Neurogliaform neurons form a novel inhibitory network in the hippocampal CA1 area. Neuroscience 25(29), 6775–6786. https://doi.org/10.1523/JNEUROSCI.1135-05.2005 (2005). Ferguson, K. A. et al. Network models provide insights into how oriens-lacunosum-moleculare and bistratified cell interactions influence the power of local hippocampal CA1 theta oscillations. Fr. Syst. Neurosci. 9, 110. https://doi.org/10.3389/fnsys.2015.00110 (2015).Allen Brain Atlas: genome-wide map of gene expression for the human adult and mouse brain ( Jones et al., 2009). Expanding a previous approach 14, where the orientation of wiring was performed through distance-based probability functions applied during pruning procedures, the PMA algorithm introduces the orientation of the probability clouds which are used directly to estimate the pairs of connections. With the present connectivity workflow, the randomization of neuronal processes is restricted to the parameter sampling procedure during network construction. It should be noted that while the pruning procedure in the PMA method is, at the moment, based on randomized sampling, in a further development of the algorithm, probabilistic parameterization based on distance could be introduced. Galindo, S. E., Toharia, P., Robles, O. D. & Pastor, L. ViSimpl: Multi-view visual analysis of brain simulation data. Fr. Neuroinform. 10, 44. https://doi.org/10.3389/fninf.2016.00044 (2016). Given the number of neurons typically included in full-scale circuits, and the specificity of morpho-anatomical constraints, network models of extended brain regions with the resolution of single cells have been limited to the olfactory bulb 30 and the striatum, where the majority of neuronal components can be assumed to be isotropically oriented 31, and a few regions of neocortex 17, cerebellum 18 and hippocampal subregions 13, 32 where strong anisotropies can be observed (e.g. cortical columns or cerebellar parallel fibers). Several brain areas show in fact a strong directionality in the connectivity patterns which need to be taken into account when building network model architecture. One typical example is the hippocampus which exhibits a peculiar anisotropic organization of neurons and axons in both position and orientation depending on the relative location of the cell soma within the hippocampal volume. The connectivity appears to be organized in such a way as to generate a preferential direction for the well-known stream of information going from the dentate gyrus to the subiculum, which has been hypothesized to be the basis for ripples and oscillatory activity 33, 34. Although different models of hippocampal neurons have been generated, ranging from detailed biophysically and morphologically accurate models 35 to advanced point-neuron integrate-and-fire implementations 36, full-scale models of hippocampal regions, such as DG, CA1, or CA3 areas based on realistic morpho-anatomical connectivity constraints are not yet available.