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Migliore, M., Cavarretta, F., Hines, M. L. & Shepherd, G. M. Distributed organization of a brain microcircuit analyzed by three-dimensional modeling: The olfactory bulb. Fr. Comput. Neurosci. 8, 50. https://doi.org/10.3389/fncom.2014.00050. (2014). de Física, Facultad de Ciencias Exactas y Naturales, Instituto de Física de Buenos Aires (IFIBA), CONICET, Universidad de Buenos Aires, Buenos Aires, Argentina Casali, S., Marenzi, E., Medini, C., Casellato, C. & D’Angelo, E. Reconstruction and simulation of a scaffold model of the cerebellar network. Front Neuroinform. 13, 37. https://doi.org/10.3389/fninf.2019.00037 (2019).

Knowles, W. D. & Schwartzkroin, P. A. Axonal ramifications of hippocampal Ca1 pyramidal cells. J. Neurosci. 1(11), 1236–1241. https://doi.org/10.1523/JNEUROSCI.01-11-01236.1981 (1981). Schneider, C. J., Cuntz, H. & Soltesz, I. Linking macroscopic with microscopic neuroanatomy using synthetic neuronal populations. PLoS Comput. Biol. 10(10), e1003921. https://doi.org/10.1371/journal.pcbi.1003921 (2014). Geminiani, A., Casellato, C., Antonietti, A., D’Angelo, E. & Pedrocchi, A. A multiple-plasticity spiking neural network embedded in a closed-loop control system to model cerebellar pathologies. Int. J. Neural Syst. 28(5), 1750017. https://doi.org/10.1142/S0129065717500174 (2018).

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Furthermore, latent ODE models can add another layer of abstraction. The observed data is assumed to be regularly/irregularly sampled from a continuous stream of data, following the dynamics described by a continuously changing hidden state. Both the dynamics of the hidden state and the relationship between the interpolated observations and the hidden state can be described by neural networks. Such systems are called Neural Control Differential Equations (Neural CDE) ( Kidger et al., 2020). Broadly speaking, they are the continuous equivalent of RNNs. 3.2.3. Differential Equations Enhanced by Deep Neural Networks In statistical physics, the mean-field approximation is a conventional way of lessening the dimensions of a many-body problem by averaging over the degrees of freedom. A well-known classic example is the problem of finding collective parameters (such as pressure or temperature) of a bulk of gas with known microscopic parameters (such as velocity or mass of the particles) by the Boltzmann distribution. The analogy of the classic gas shows the gist of the neural mass model: the temperature is an emergent phenomenon of the gas ensemble. Although higher temperatures correspond to higher average velocity of the particles, one needs a computational bridge to map microscopic parameters to the macroscopic one(s). To be clear, remember that each particle has many relevant attributes (e.g., velocity, mass, and the interaction force relative to other particles). Each attribute denotes one dimension in the phase space. One can immediately see how this problem can become computationally impossible even for 1 cm 3 of gas with ~10 19 molecules. De Schepper, R. et al. Scaffold modelling captures the structure–function–dynamics relationship in brain microcircuits. bioRxiv https://doi.org/10.1101/2021.07.30.454314 (2021).

Diesmann, M., Gewaltig, M. O. & Aertsen, A. Stable propagation of synchronous spiking in cortical neural networks. Nature 402, 529–533 (1999). Kitamura, K., Judkewitz, B., Kano, M. & Häusser, M. Targeted patch-clamp recordings and single-cell electroporation of unlabeled neurons in vivo. Nat. Methods 5(1), 61–67. https://doi.org/10.1038/nmeth1150 (2008). In this scenario, the advent of High-Performance Computing, coupled with a huge amount of experimental data, boosted the development of extended data-driven spiking neural network models 16, 17, 18, 19. Irrespective of the single cell model employed and the level of electrophysiological detail, the connectivity strategy remains a critical determinant in the construction of networks 20. Schneider, C. J., Bezaire, M. & Soltesz, I. Towards a full-scale computational model of the rat dentate gyrus. Fr. Neural Circuits. 6, 83. https://doi.org/10.3389/fncir.2012.00083 (2012). 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.RNNs vary greatly in architecture. The choice of architecture can be implied by the output of interest (for example text Sutskever et al., 2011 vs. natural scenes Socher et al., 2011) or the approaches to overcome the problem with vanishing and exploding gradient (e.g., long short-term memory (LSTM) Hochreiter and Schmidhuber, 1997, hierarchical Hihi and Bengio, 1995, or gated RNNs Chung et al., 2014). 3.1.2.1. Hopfield Zeng, Y. et al. Understanding the impact of neural variations and random connections on inference. Front. Comput. Neurosci. 15, 612937. https://doi.org/10.3389/fncom.2021.612937 (2021). Bezaire, M. J., Raikov, I., Burk, K., Vyas, D. & Soltesz, I. Interneuronal mechanisms of hippocampal theta oscillations in a full-scale model of the rodent CA1 circuit. Elife 5, e18566. https://doi.org/10.7554/eLife.18566 (2016). Giacopelli, G., Tegolo, D., Spera, E. & Migliore, M. On the structural connectivity of large-scale models of brain networks at cellular level. Sci. Rep. 11(1), 4345. https://doi.org/10.1038/s41598-021-83759-z (2021). Recurrent neural networks (RNN) are the Turing-complete ( Kilian and Siegelmann, 1996) algorithms for learning dynamics and are widely used in computational neuroscience. In a nutshell, RNN processes data by updating a “state vector.” The state vector holds the memory across steps in the sequence. This state vector contains long-term information of the sequence from the past steps ( LeCun et al., 2015).

Biophysical models: Biophysical models are realistic models which encapsulate biological assumptions and constraints. Due to large number of components and the empirical complexity of the systems modeled, examples of biophysical models run the gamut, from very small, with a high degree of realism (e.g., Hodgkin and Huxley's model of squid giant axon), to large scale (e.g., Izhikevich and Edelman, 2008 model of whole cortex). Due to computational limitations, large-scale models are often accompanied by increasing levels of simplification. Blue Brain Project ( Markram, 2006) is an example of this type of modeling. The Perforant pathway associated cells have their cell bodies in the SR and their axonal extents are confined in the SR and SLM while their dendrites project in both directions 39. We have modeled the dendrites as two cones with vertex on the soma and a large ellipsoid reproducing the axonal plexus (Fig. 5). DCM can be thought of as a method of finding the optimal parameters of the causal relations that best fit the observed data. The parameters of the connectivity network are (1) anatomical and functional couplings, (2) induced effect of stimuli, and (3) the parameters that describe the influence of the input on the intrinsic couplings. The expectation-maximization (EM) algorithm is the widely-used optimizer. However, EM is slow for large, changing, and/or noisy networks. Zhuang et al. (2021) showed Multiple-Shooting Adjoint Method for Whole-Brain Dynamic outperforming EM on classification tasks while being used for continuous and changing networks. A notable effort in this regard is the Allen Brain Atlas ( Jones et al., 2009) in which genomic data of mice, humans, and non-human primates ( Hawrylycz et al., 2014) have been collected and mapped for understanding structural and functional architecture of the brain ( Gilbert, 2018). While genomic data by itself is valuable for mapping out connectivity in different cell types, a fifth division of AA, Allen Institute for Neural Dynamics was recently announced, with the aim of studying the link between neural circuits of laboratory mice and behaviors related to foraging ( Chen and Miller, 2021).Madisen, L. et al. A robust and high-throughput Cre reporting and characterization system for the whole mouse brain. Nat. Neurosci. 13(1), 133–140. https://doi.org/10.1038/nn.2467 (2010). Cutsuridis, V., Cobb, S. & Graham, B. P. Encoding and retrieval in a model of the hippocampal CA1 microcircuit. Hippocampus 20(3), 423–446. https://doi.org/10.1002/hipo.20661 (2010). According to the heterogeneity of shapes and orientations of inhibitory interneurons, we have identified 11 classes of cells which were grouped into 7 different shapes generated through combinations of axonal and dendritic probability: Breakspear, M. Dynamic models of large-scale brain activity. Nat. Neurosci. 20(3), 340–352. https://doi.org/10.1038/nn.4497 (2017).

The following sections describe general function approximators that could identify data dynamics without injecting any prior knowledge about the system. They could provide a perfect solution for a well-observed system with unknown dynamics. Although some of neural ODE methods have already been applied to fMRI and EEG data ( Zhuang et al., 2021), other deep architectures such as GOKU-net and latent ODEs are new frontiers. 3.2.1. Sparse Identification of Nonlinear Dynamics Put simply: the goal of science is to leverage prior knowledge, not merely to forecast the future (a task well suited to engineering problems), but to answer “why,” questions, and to facilitate the discovery of mechanisms and principles of operation. Bzdok and Ioannidis (2019) discuss why inference should be prioritized over prediction for building a reproducible and expandable body of knowledge. We argue that this priority should be especially respected for clinical neuroscience.

Tsodysk, M. V. & Markram, H. The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability. PNAS 94(2), 719–723. https://doi.org/10.1073/pnas.94.2.719 (1997). Ferrante, M., Migliore, M. & Ascoli, G. A. Feed-forward inhibition as a buffer of the neuronal input-output relation. Proc. Natl. Acad. Sci. U S A 106(42), 18004–18009. https://doi.org/10.1073/pnas.0904784106 (2009). 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