Leaders of neuronal cultures in a quorum percolation model

The vitals for steady nucleation maps of spontaneous spiking coherence in autonomous two-dimensional neuronal networks

Thin pancake-like neuronal networks cultured on top of a planar microelectrode array have been extensively tried out in neuroengineering, as a substrate for the mobile robot's control unit, i.e., as a cyborg's brain. Most of these attempts failed due to intricate self-organizing dynamics in the neuronal systems. In particular, the networks may exhibit an emergent spatial map of steady nucleation sites ("n-sites") of spontaneous population spikes. Being unpredictable and independent of the surface electrode locations, the n-sites drastically change local ability of the network to generate spikes. Here, using a spiking neuronal network model with generative spatially-embedded connectome, we systematically show in simulations that the number, location, and relative activity of spontaneously formed n-sites ("the vitals") crucially depend on the samplings of three distributions: (1) the network distribution of neuronal excitability, (2) the distribution of connections between neurons of the network, and (3) the distribution of maximal amplitudes of a single synaptic current pulse. Moreover, blocking the dynamics of a small fraction (about 4%) of non-pacemaker neurons having the highest excitability was enough to completely suppress the occurrence of population spikes and their n-sites. This key result is explained theoretically. Remarkably, the n-sites occur taking into account only short-term synaptic plasticity, i.e., without a Hebbian-type plasticity. As the spiking network model used in this study is strictly deterministic, all simulation results can be accurately reproduced. The model, which has already demonstrated a very high richness-to-complexity ratio, can also be directly extended into the three-dimensional case, e.g., for targeting peculiarities of spiking dynamics in cerebral (or brain) organoids. We recommend the model as an excellent illustrative tool for teaching network-level computational neuroscience, complementing a few benchmark models.

Influence of microgravity on spontaneous calcium activity of primary hippocampal neurons grown in microfluidic chips

The influence of variations of gravity, either hypergravity or microgravity, on the brain of astronauts is a major concern for long journeys in space, to the Moon or to Mars, or simply long-duration missions on the ISS (International Space Station). Monitoring brain activity, before and after ISS missions already demonstrated important and long term effects on the brains of astronauts. In this study, we focus on the influence of gravity variations at the cellular level on primary hippocampal neurons. A dedicated setup has been designed and built to perform live calcium imaging during parabolic flights. During a CNES (Centre National d'Etudes Spatiales) parabolic flight campaign, we were able to observe and monitor the calcium activity of 2D networks of neurons inside microfluidic devices during gravity changes over different parabolas. Our preliminary results clearly indicate a modification of the calcium activity associated to variations of gravity.

Flexibility of in vitro cortical circuits influences resilience from microtrauma

Background

Small clusters comprising hundreds to thousands of neurons are an important level of brain architecture that correlates single neuronal properties to fulfill brain function, but the specific mechanisms through which this scaling occurs are not well understood. In this study, we developed an in vitro experimental platform of small neuronal circuits (islands) to probe the importance of structural properties for their development, physiology, and response to microtrauma.

Methods

Primary cortical neurons were plated on a substrate patterned to promote attachment in clusters of hundreds of cells (islands), transduced with GCaMP6f, allowed to mature until 10-13 days in vitro (DIV), and monitored with Ca²⁺ as a non-invasive proxy for electrical activity. We adjusted two structural factors-island size and cellular density-to evaluate their role in guiding spontaneous activity and network formation in neuronal islands.

Results

We found cellular density, but not island size, regulates of circuit activity and network function in this system. Low cellular density islands can achieve many states of activity, while high cellular density biases islands towards a limited regime characterized by low rates of activity and high synchronization, a property we summarized as "flexibility." The injury severity required for an island to lose activity in 50% of its population was significantly higher in low-density, high flexibility islands.

Conclusion

Together, these studies demonstrate flexible living cortical circuits are more resilient to microtrauma, providing the first evidence that initial circuit state may be a key factor to consider when evaluating the consequences of trauma to the cortex.

Stimulation triggers endogenous activity patterns in cultured cortical networks

Cultures of dissociated cortical neurons represent a powerful trade-off between more realistic experimental models and abstract modeling approaches, allowing to investigate mechanisms of synchronized activity generation. These networks spontaneously alternate periods of high activity (i.e. network bursts) with periods of quiescence in a dynamic state which recalls the fluctuation of in vivo UP and DOWN states. Network bursts can also be elicited by external stimulation and their spatial propagation patterns tracked by means of multi-channel micro-electrode arrays. In this study, we used rat cortical cultures coupled to micro-electrode arrays to investigate the similarity between spontaneous and evoked activity patterns. We performed experiments by applying electrical stimulation to different network locations and demonstrated that the rank orders of electrodes during evoked and spontaneous events are remarkably similar independently from the stimulation source. We linked this result to the capability of stimulation to evoke firing in highly active and “leader” sites of the network, reliably and rapidly recruited within both spontaneous and evoked bursts. Our study provides the first evidence that spontaneous and evoked activity similarity is reliably observed also in dissociated cortical networks.

Chimera-like states in structured heterogeneous networks

Chimera-like states are manifested through the coexistence of synchronous and asynchronous dynamics and have been observed in various systems. To analyze the role of network topology in giving rise to chimera-like states, we study a heterogeneous network model comprising two groups of nodes, of high and low degrees of connectivity. The architecture facilitates the analysis of the system, which separates into a densely connected coherent group of nodes, perturbed by their sparsely connected drifting neighbors. It describes a synchronous behavior of the densely connected group and scaling properties of the induced perturbations.

A spatially resolved network spike in model neuronal cultures reveals nucleation centers, circular traveling waves and drifting spiral waves

We show that in model neuronal cultures, where the probability of interneuronal connection formation decreases exponentially with increasing distance between the neurons, there exists a small number of spatial nucleation centers of a network spike, from where the synchronous spiking activity starts propagating in the network typically in the form of circular traveling waves. The number of nucleation centers and their spatial locations are unique and unchanged for a given realization of neuronal network but are different for different networks. In contrast, if the probability of interneuronal connection formation is independent of the distance between neurons, then the nucleation centers do not arise and the synchronization of spiking activity during a network spike occurs spatially uniform throughout the network. Therefore one can conclude that spatial proximity of connections between neurons is important for the formation of nucleation centers. It is also shown that fluctuations of the spatial density of neurons at their random homogeneous distribution typical for the experiments in vitro do not determine the locations of the nucleation centers. The simulation results are qualitatively consistent with the experimental observations.

Effect of threshold disorder on the quorum percolation model

We study the modifications induced in the behavior of the quorum percolation model on neural networks with Gaussian in-degree by taking into account an uncorrelated Gaussian thresholds variability. We derive a mean-field approach and show its relevance by carrying out explicit Monte Carlo simulations. It turns out that such a disorder shifts the position of the percolation transition, impacts the size of the giant cluster, and can even destroy the transition. Moreover, we highlight the occurrence of disorder independent fixed points above the quorum critical value. The mean-field approach enables us to interpret these effects in terms of activation probability. A finite-size analysis enables us to show that the order parameter is weakly self-averaging with an exponent independent on the thresholds disorder. Last, we show that the effects of the thresholds and connectivity disorders cannot be easily discriminated from the measured averaged physical quantities.

Finite-size effects and dynamics of giant transition of a continuum quorum percolation model on random networks

We start from a continuous extension of a mean field approach of the quorum percolation model, accounting for the response of in vitro neuronal cultures, to carry out a normal form analysis of the critical behavior. We highlight the effects of nonlinearities associated with this mean field approach even in the close vicinity of the critical point. Statistical properties of random networks with Gaussian in-degree are related to the outcoming links distribution. Finite size analysis of explicit Monte Carlo simulations enables us to confirm the relevance of the mean field approach on such networks and to show that the order parameter is weakly self-averaging; dynamical relaxation is investigated. Furthermore we derive a mean field equation taking into account the effect of inhibitory neurons and discuss the equivalence with a purely excitatory network.

Self-referring DNA and protein: a remark on physical and geometrical aspects

All known life forms are based upon a hierarchy of interwoven feedback loops, operating over a cascade of space, time and energy scales. Among the most basic loops are those connecting DNA and proteins. For example, in genetic networks, DNA genes are expressed as proteins, which may bind near the same genes and thereby control their own expression. In this molecular type of self-reference, information is mapped from the DNA sequence to the protein and back to DNA. There is a variety of dynamic DNA-protein self-reference loops, and the purpose of this remark is to discuss certain geometrical and physical aspects related to the back and forth mapping between DNA and proteins. The mappings are examined as dimensional reductions and expansions between high- and low-dimensional manifolds in molecular spaces. The discussion raises basic questions regarding the nature of DNA and proteins as self-referring matter, which are examined in a simple toy model.

Functional Clusters, Hubs, and Communities in the Cortical Microconnectome

Although relationships between networks of different scales have been observed in macroscopic brain studies, relationships between structures of different scales in networks of neurons are unknown. To address this, we recorded from up to 500 neurons simultaneously from slice cultures of rodent somatosensory cortex. We then measured directed effective networks with transfer entropy, previously validated in simulated cortical networks. These effective networks enabled us to evaluate distinctive nonrandom structures of connectivity at 2 different scales. We have 4 main findings. First, at the scale of 3-6 neurons (clusters), we found that high numbers of connections occurred significantly more often than expected by chance. Second, the distribution of the number of connections per neuron (degree distribution) had a long tail, indicating that the network contained distinctively high-degree neurons, or hubs. Third, at the scale of tens to hundreds of neurons, we typically found 2-3 significantly large communities. Finally, we demonstrated that communities were relatively more robust than clusters against shuffling of connections. We conclude the microconnectome of the cortex has specific organization at different scales, as revealed by differences in robustness. We suggest that this information will help us to understand how the microconnectome is robust against damage.

Random graph theory and neuropercolation for modeling brain oscillations at criticality

Mathematical approaches are reviewed to interpret intermittent singular space-time dynamics observed in brain imaging experiments. The following aspects of brain dynamics are considered: nonlinear dynamics (chaos), phase transitions, and criticality. Probabilistic cellular automata and random graph models are described, which develop equations for the probability distributions of macroscopic state variables as an alternative to differential equations. The introduced modular neuropercolation model is motivated by the multilayer structure and dynamical properties of the cortex, and it describes critical brain oscillations, including background activity, narrow-band oscillations in excitatory-inhibitory populations, and broadband oscillations in the cortex. Input-induced and spontaneous transitions between states with large-scale synchrony and without synchrony exhibit brief episodes with long-range spatial correlations as observed in experiments.

Transfer entropy reconstruction and labeling of neuronal connections from simulated calcium imaging

Neuronal dynamics are fundamentally constrained by the underlying structural network architecture, yet much of the details of this synaptic connectivity are still unknown even in neuronal cultures in vitro. Here we extend a previous approach based on information theory, the Generalized Transfer Entropy, to the reconstruction of connectivity of simulated neuronal networks of both excitatory and inhibitory neurons. We show that, due to the model-free nature of the developed measure, both kinds of connections can be reliably inferred if the average firing rate between synchronous burst events exceeds a small minimum frequency. Furthermore, we suggest, based on systematic simulations, that even lower spontaneous inter-burst rates could be raised to meet the requirements of our reconstruction algorithm by applying a weak spatially homogeneous stimulation to the entire network. By combining multiple recordings of the same in silico network before and after pharmacologically blocking inhibitory synaptic transmission, we show then how it becomes possible to infer with high confidence the excitatory or inhibitory nature of each individual neuron.

Model-free reconstruction of excitatory neuronal connectivity from calcium imaging signals

A systematic assessment of global neural network connectivity through direct electrophysiological assays has remained technically infeasible, even in simpler systems like dissociated neuronal cultures. We introduce an improved algorithmic approach based on Transfer Entropy to reconstruct structural connectivity from network activity monitored through calcium imaging. We focus in this study on the inference of excitatory synaptic links. Based on information theory, our method requires no prior assumptions on the statistics of neuronal firing and neuronal connections. The performance of our algorithm is benchmarked on surrogate time series of calcium fluorescence generated by the simulated dynamics of a network with known ground-truth topology. We find that the functional network topology revealed by Transfer Entropy depends qualitatively on the time-dependent dynamic state of the network (bursting or non-bursting). Thus by conditioning with respect to the global mean activity, we improve the performance of our method. This allows us to focus the analysis to specific dynamical regimes of the network in which the inferred functional connectivity is shaped by monosynaptic excitatory connections, rather than by collective synchrony. Our method can discriminate between actual causal influences between neurons and spurious non-causal correlations due to light scattering artifacts, which inherently affect the quality of fluorescence imaging. Compared to other reconstruction strategies such as cross-correlation or Granger Causality methods, our method based on improved Transfer Entropy is remarkably more accurate. In particular, it provides a good estimation of the excitatory network clustering coefficient, allowing for discrimination between weakly and strongly clustered topologies. Finally, we demonstrate the applicability of our method to analyses of real recordings of in vitro disinhibited cortical cultures where we suggest that excitatory connections are characterized by an elevated level of clustering compared to a random graph (although not extreme) and can be markedly non-local.

Unraveling the general organizing principles of connectivity in neural circuits is a crucial step towards understanding brain function. However, even the simpler task of assessing the global excitatory connectivity of a culture in vitro, where neurons form self-organized networks in absence of external stimuli, remains challenging. Neuronal cultures undergo spontaneous switching between episodes of synchronous bursting and quieter inter-burst periods. We introduce here a novel algorithm which aims at inferring the connectivity of neuronal cultures from calcium fluorescence recordings of their network dynamics. To achieve this goal, we develop a suitable generalization of Transfer Entropy, an information-theoretic measure of causal influences between time series. Unlike previous algorithmic approaches to reconstruction, Transfer Entropy is data-driven and does not rely on specific assumptions about neuronal firing statistics or network topology. We generate simulated calcium signals from networks with controlled ground-truth topology and purely excitatory interactions and show that, by restricting the analysis to inter-bursts periods, Transfer Entropy robustly achieves a good reconstruction performance for disparate network connectivities. Finally, we apply our method to real data and find evidence of non-random features in cultured networks, such as the existence of highly connected hub excitatory neurons and of an elevated (but not extreme) level of clustering.

Interactions between network synchrony and the dynamics of neuronal threshold

Synchronous activity impacts on a range of functional brain capacities in health and disease. To address the interrelations between cellular level activity and network-wide synchronous events, we implemented in vitro a recently introduced technique, the response clamp, which enables online monitoring of single neuron threshold dynamics while ongoing network synchronous activity continues uninterrupted. We show that the occurrence of a synchronous network event causes a significant biphasic change in the single neuron threshold. These threshold dynamics are correlated across the neurons constituting the network and are entailed by the input to the neurons rather than by their own spiking (i.e., output) activity. The magnitude of network activity during a synchronous event is correlated with the threshold state of individual neurons at the event's onset. Recovery from the impact of a given synchronous event on the neuronal threshold lasts several seconds and seems to be a key determinant of the time to the next spontaneously occurring synchronous event. Moreover, the neuronal threshold is shown to be correlated with the excitability dynamics of the entire network. We conclude that the relations between the two levels (network activity and the single neuron threshold) should be thought of in terms that emphasize their interactive nature.

The emergence of connectivity in neuronal networks: from bootstrap percolation to auto-associative memory

We consider a random synaptic pruning in an initially highly interconnected network. It is proved that a random network can maintain a self-sustained activity level for some parameters. For such a set of parameters a pruning is constructed so that in the resulting network each neuron/node has almost equal numbers of in- and out-connections. It is also shown that the set of parameters which admits a self-sustained activity level is rather small within the whole space of possible parameters. It is pointed out here that the threshold of connectivity for an auto-associative memory in a Hopfield model on a random graph coincides with the threshold for the bootstrap percolation on the same random graph. It is argued that this coincidence reflects the relations between the auto-associative memory mechanism and the properties of the underlying random network structure. This article is part of a Special Issue entitled "Neural Coding".

Propagation of action potential activity in a predefined microtunnel neural network

A polydimethylsiloxane microtunnel device with two wells is aligned and attached on top of a multi-electrode array. Neurons are grown first in one well and allow the propagation of axons through the tunnels into a second well. After 10 days, cells are plated in the second well, with much lower likelihood of extending axons back to the first well, with the intent of creating unidirectional connectivity between populations of neurons in the two wells. Here we report electrophysiological evidence that supports the hypothesis that the dominant information flow is in the desired direction. This was done by measuring the propagation speed and direction of individual action potentials, with the result that 84% of the spikes propagated in the desired direction. Further, we recorded globally synchronized burst activity on each of the electrodes, identified the timing of the first spike on each electrode, recorded locally synchronized burst activity which is found only in the second well and does not propagate back to the first well and concluded that this measure of burst propagation supports the hypothesis of a unidirectionally connected network. Two hypotheses are discussed for the mechanism underlying the activity pattern of the particular neural networks.

Representation and learning in neuronal networks: a conceptual nervous system approach

The work presented in this review describes the use of large cortical networks developing ex vivo, in a culture dish, to study principles underlying synchronization, adaptation, learning, and representation in neuronal assemblies. The motivation to study neuronal networks ex vivo is outlined together with a short description of recent results in this field. Following a short description of the experimental system, a set of basic results will be presented that concern self-organization of activity, dynamical and functional properties of neurons and networks in response to external stimulation. This short review ends with an outline of future questions and research directions.