Neuromedic and Clinical Neurophysiology and Neuromathemathical Models Research

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Mentirosa, manipuladora, miserable, mezquina, perversa… por eso están en la basura opositora!

Conectando los conectomas con la fisiología
mayo 2023The Journal of Neuroscience: El Diario Oficial de la Sociedad de Neurociencia 43(20):3599-3610
DOI:10.1523/JNEUROSCI.2208-22.2023
LicenciaCC BY-NC-SA 4.0
Autores:
Alejandro Borst
Christian Leibold
Universidad de Friburgo

Sincronicidad de las concentraciones circulantes de leptina, hormona luteinizante y estradiol de muestras frecuentes durante 24 h en mujeres sanas
marzo de 1998Actas de la Academia Nacional de Ciencias 95(5):2541-2546
DOI:10.1073/pnas.95.5.2541
FuentePubMed

Mec´anica de Medios Continuos:Resumen de ´Algebra y C´alculoTensorial
Jos´e M.
a
Goicolea Ruig´omez,
Depto. de Mec´anica de Medios Continuos y Teor´ıa de Estructuras,Universidad Polit´ecnica de Madrid
8 de octubre, 2002

The structure and function of neural
connectomes are shaped by a small number of
design principles
Adam Haber1, Adrian A. Wanner2, Rainer W. Friedrich3,4, and Elad Schneidman1,
1
Department of Brain Sciences, Weizmann Institute of Science, Rehovot, Israel 2
Paul Scherrer Institute, Villigen, Switzerland 3
Friedrich Miescher Institute for Biomedical Research, Basel, Switzerland 4
University of Basel, Basel, Switzerland
The map of synaptic connectivity among neurons in the brain
shapes the computations that neural circuits may perform. Inferring the design principles of neural connectomes is, therefore, fundamental for understanding brain development and architecture, neural computations, learning, and behavior. Here,
we learn probabilistic generative models for the connectomes of
the olfactory bulb of zebrafish, part of the mouse visual cortex,
and of C. elegans. We show that, in all cases, models that rely
on a surprisingly small number of simple biological and physical features are highly accurate in replicating a wide range of
properties of the measured circuits. Specifically, they accurately
predict the existence of individual synapses and their strength,
distributions of synaptic indegree and outdegree of the neurons,
frequency of sub-network motifs, and more. Furthermore, we
simulate synthetic circuits generated by our model for the olfactory bulb of zebrafish and show that they replicate the computation that the real circuit performs in response to olfactory
cues. Finally, we show that specific failures of our models reflect
missing design features that we uncover by adding latent features to the model. Thus, our results reflect surprisingly simple
design principles of real connectomes in three different systems
and species, and offer a novel general computational framework
for analyzing connectomes and linking structure and function in
neural circuits.
Connectomes | Neural Networks | Generative Models | Maximum Entropy Models | Network Structure and Function | Graph Architecture
Correspondence: [email protected]
Introduction
The ability to reconstruct the detailed connectivity maps between neurons at unprecedented scale and resolution (1–5)
opens the door for the quantitative analysis of the organization of real neural networks, the inference of their structural design principles, and the uncovering of the relations between structure and function in neural circuits (6–8). Analyses of the detailed architecture of brain networks have shown
that synaptic connectivity is structured at different scales:
from the enrichment of reciprocal connections (9) and network motifs (10–12), through the structured organization of
the retina (13), to cortical columns (14–16), and other nervous systems and brain areas (1, 17–19). A wide range of genetic (20–23), morphological (24, 25), biochemical (26), and
economical (27) mechanisms have been implied or shown to
play a significant role in shaping the structure of these networks, but the interplay between these mechanisms is still
not well understood. Moreover, it remains unclear to what
extent the large-scale organization of brain networks can be
explained by simple and local connectivity rules. Complementary to this “bottom-up” approach, network theory tools
(28, 29) have been used to study the design of networks of
neurons (30, 31), yielding models able to reproduce global
properties, such as “small-worldness" (32) and modular organizations (33). However, it is unclear how these properties
would emerge from local synaptic formation mechanisms,
such as the ones described above. Here, we merge these two
viewpoints of network analysis and design by learning generative models of large connectomes whose basic building
blocks correspond to simple and natural biological structural
features and physical constraints.
Inferring and understanding the the design principles of networks and their organization require a modeling formalism
that would allow us to bring together local rules, global network structures, and the inherent stochastic nature of individual networks. This is because any given connectome is
the result of genetic instructions, developmental processes,
and learning throughout the lifetime of the organism – which
are all noisy biophysical processes that also depend on other
external conditions. It is clear then that the detailed connectivity of the same areas across animals would be probabilistic
in nature, and no two circuits would have exactly the same
neurons or connections (except, maybe, for specialized circuits with “identified" neurons). So, to describe and evaluate connectomes, we must rely on statistical models over the
possible connectivity maps. Such maps of connections between neurons are naturally described as a directed graph or
a matrix G, where each entry Gij reflects the synaptic connections from neuron i to neuron j. A statistical model of
connectomes would then be a probability distribution over
connectivity matrices, G, that depends on a set of parameters
◊.
Learning such accurate models for connectomes, P◊(G),
would allow us to (1) evaluate the likelihood of a particular
connectivity map, (2) generate synthetic connectomes that resemble the observed ones, by sampling from the model, (3)
quantify the expected values of different features of the connectome and their variability, and (4) explore the functional
design of connectomes, by simulating the neural dynamics
Haber et al. | bioR‰iv | March 15, 2023 | 1–19
available under aCC-BY-NC-ND 4.0 International license.
(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
bioRxiv preprint doi: https://doi.org/10.1101/2023.03.15.532611; this version posted March 15, 2023. The copyright holder for this preprint
of the sampled networks. From a computational perspective,
we would seek models that recapitulate the observed structure as accurately as possible, and identify the minimal set
of features that would be needed. From a biological perspective, we would like to find models that rely on biologicallyplausible or “understandable" features. Learning such statistical models is a challenging computational task,to which
should be added in this study the hurdle of having only a
few examples of reconstructed connectomes. Consequently,
we aim to learn models based on the observed values of a
small number of simple and biologically-plausible features
or statistics of the connectome, as summarized in Fig. 1, and
evaluate these models using a range of statistical and structural measures that we did not use for learning the models –
at the level of individual synapses, individual neurons, small
sub-circuits, and the circuit as a whole. We then ask how
well these models predict the function of the circuit, what
they might reveal about the design principles that we are still
missing, and how we might find them.
Results
Learning accurate generative models of connectomes
based on simple biological and physical features. We
first consider the connectome of part of the olfactory bulb
(OB) of a zebrafish larva, taken from (34). This reconstructed
network comprises more than 40% of the entire OB, containing 208 excitatory mitral cells (MCs) and 238 inhibitory interneurons (INs) that are connected by 9919 synapses (out of
198,470 possible ones), and the activity of its principal neurons has been measured (as we will see below). We start by
using the unweighted version of the connectome (Fig. 2a; see
Methods), as the estimation of synaptic strengths from structural data is sometimes partial or noisy (35). The connectome
in this case corresponds to a binary matrix, G, where Gij = 1
designates the existence of a synapse or multiple synapses
from neuron i to neuron j.
To identify the design principles of the OB connectome, we
asked how well would models that rely on simple structural and biophysical features successfully replicate the observed connectivity. The simplest connectome model is one
that relies solely on one structural feature – the total number of synapses in the network – and would adjust the overall sparsity of synaptic connections to match the observed
one (4.9%). This is the well-known Erdos-Re ˝ nyi (ER) ran- ´
dom graph model (28), which assigns the same probability to
all potential synapses between neurons in the network. Because of its inherent homogeneity, this model gives a probability map for synaptic connections (i.e., a matrix containing
the probabilities of synaptic connections between all pairs of
neurons) that does not show any structure, (Supplementary
Fig. S1), and is clearly a poor model of the real connectome .
We then learn eight models for the connectome, that in addition to retaining the total number of synapses that were
observed in the data, each model relies on a different set of
features, and the model parameters are learned so that the
expected feature values are consistent with their measured
ones. Each of the eight feature-sets has a clear biological
or physical interpretation: (1) cell-type-specific connectivity, (2) distance-dependent connectivity between neurons, (3)
reciprocity of connections between pairs of neurons, (4) the
dependence of incoming synapses on the location of the postsynaptic neuron, (5) the dependence of outgoing synapses
on the spatial locations of the presynaptic neuron, (6) preferential attachment between Glomeruli, (7) the effect of dendritic tree sizes on the number of incoming synapses, and (8)
the effect of dendritic tree sizes on the number of outgoing
synapses (see Methods). Thus, for example, the model that
relies on cell-type specific connectivity has 4 specific features
– corresponding to the 2◊2 values of the probability of a neuron of one type to have a synapse with a neuron of another
type (as the neurons in this data set were identified as either
MCs or INs). To assess the effect of each feature-set, we learn
in each case the most random or least structured probabilistic
model over networks, P◊(G), which matches the observed
values of the corresponding features. Thus, given a set of
features {fµ(G)}, we find the maximum entropy distribution
over networks such that the average values of these features
over the model and over the given connectome agree, namely,
Èfµ(G)ÍP = Èfµ(G)Ídata. This model is given by
P◊(G) = 1
Z exp{≠
ÿ
µ
◊µfµ(G)},
where the ׵s are found numerically and Z is a normalization term or partition function (see Methods). These models,
also known as Exponential Random Graph Models (36), not
only allow us to compute the likelihood of any given connectome, but also serve as generative models that can be used to
sample connectomes that are consistent with their respective
features. Importantly, because these are maximum entropy
models (37), the solution is unique and assumes no other
structure beyond the measured features. These models, therefore, capture the full predictive power of their respective set
of features. Fig. 2b shows the probability maps over the full
connectivity matrix of synapses for each of the these models.
As is evident, they each capture some aspects of the connectivity matrix, but none of them is particularly accurate on its
own.
The generality of this mathematical framework means that
we can naturally extend it to learn a model for the connectome that relies on any combination of the different featuresets simultaneously (as in (38)). Fig. 2c shows the synaptic probability map of this model and its similarity to the
data (shown in Fig. 2a). Importantly, while predictions of
the model may reflect high variance or uncertainty regarding
individual synapses, existing synapses were assigned higher
probabilities, despite the fact that the model was only trained
on the aggregate summary statistics of the connectivity matrix (Fig. 2d).
We next quantify the performance of the different models
by their ability to predict individual synapses, as well as
the connectivity profiles of individual neurons, sub-circuit
properties, and the likelihood of the connectivity map of
the whole circuit. We first asked how accurately they predict the existence of individual synapses. This was assessed
2 | bioR‰iv Hab