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Timescales of neural activity are diverse across and within brain areas, and experimental observations suggest that neural timescales reflect information in dynamic environments. However, these observations do not specify how neural timescales are shaped, nor whether particular timescales are necessary for neural computations and brain function. Here, we take a complementary perspective and synthesize three directions where computational methods can distill the broad set of empirical observations into quantitative and testable theories: We review (i) how data analysis methods allow us to capture different timescales of neural dynamics across different recording modalities, (ii) how computational models provide a mechanistic explanation for the emergence of diverse timescales, and (iii) how task-optimized models in machine learning uncover the functional relevance of neural timescales. This integrative computational approach, combined with empirical findings, would provide a more holistic understanding of how neural timescales capture the relationship between brain structure, dynamics, and behavior., Comment: 18 pages, 4 figures, 2 boxes

Structural modularity is a pervasive feature of biological neural networks, which have been linked to several functional and computational advantages. Yet, the use of modular architectures in artificial neural networks has been relatively limited despite early successes. Here, we explore the performance and functional dynamics of a modular network trained on a memory task via an iterative growth curriculum. We find that for a given classical, non-modular recurrent neural network (RNN), an equivalent modular network will perform better across multiple metrics, including training time, generalizability, and robustness to some perturbations. We further examine how different aspects of a modular network's connectivity contribute to its computational capability. We then demonstrate that the inductive bias introduced by the modular topology is strong enough for the network to perform well even when the connectivity within modules is fixed and only the connections between modules are trained. Our findings suggest that gradual modular growth of RNNs could provide advantages for learning increasingly complex tasks on evolutionary timescales, and help build more scalable and compressible artificial networks.

Many settings in machine learning require the selection of a rotation representation. However, choosing a suitable representation from the many available options is challenging. This paper acts as a survey and guide through rotation representations. We walk through their properties that harm or benefit deep learning with gradient-based optimization. By consolidating insights from rotation-based learning, we provide a comprehensive overview of learning functions with rotation representations. We provide guidance on selecting representations based on whether rotations are in the model's input or output and whether the data primarily comprises small angles., Comment: Published at ICML 2024

Developing reliable mechanisms for continuous local learning is a central challenge faced by biological and artificial systems. Yet, how the environmental factors and structural constraints on the learning network influence the optimal plasticity mechanisms remains obscure even for simple settings. To elucidate these dependencies, we study meta-learning via evolutionary optimization of simple reward-modulated plasticity rules in embodied agents solving a foraging task. We show that unconstrained meta-learning leads to the emergence of diverse plasticity rules. However, regularization and bottlenecks to the model help reduce this variability, resulting in interpretable rules. Our findings indicate that the meta-learning of plasticity rules is very sensitive to various parameters, with this sensitivity possibly reflected in the learning rules found in biological networks. When included in models, these dependencies can be used to discover potential objective functions and details of biological learning via comparisons with experimental observations.

As more connectome data become available, the question of how to best analyse the structure of biological neural networks becomes increasingly pertinent. In brain networks, knowing that two areas are connected is often not sufficient, as the directionality and weight of the connection affect the dynamics in crucial ways. Still, the methods commonly used to estimate network properties, such as clustering and small-worldness, usually disregard features encoded in the directionality and strength of network connections. To address this issue, we propose using fully-weighted and directed clustering measures that provide higher sensitivity to non-random structural features. Using artificial networks, we demonstrate the problems with methods routinely used in the field and how fully-weighted and directed methods can alleviate them. Specifically, we highlight their robustness to noise and their ability to address thresholding issues, particularly in inferred networks. We further apply our method to the connectomes of different species and uncover regularities and correlations between neuronal structures and functions that cannot be detected with traditional clustering metrics. Finally, we extend the notion of small-worldness in brain networks to account for weights and directionality and show that some connectomes can no longer be considered ``small-world''. Overall, our study makes a case for a combined use of fully-weighted and directed measures to deal with the variability of brain networks and suggests the presence of complex patterns in neural connectivity that can only be revealed using such methods.

Recurrent neural networks (RNNs) in the brain and in silico excel at solving tasks with intricate temporal dependencies. Long timescales required for solving such tasks can arise from properties of individual neurons (single-neuron timescale, $\tau$, e.g., membrane time constant in biological neurons) or recurrent interactions among them (network-mediated timescale). However, the contribution of each mechanism for optimally solving memory-dependent tasks remains poorly understood. Here, we train RNNs to solve $N$-parity and $N$-delayed match-to-sample tasks with increasing memory requirements controlled by $N$ by simultaneously optimizing recurrent weights and $\tau$s. We find that for both tasks RNNs develop longer timescales with increasing $N$, but depending on the learning objective, they use different mechanisms. Two distinct curricula define learning objectives: sequential learning of a single-$N$ (single-head) or simultaneous learning of multiple $N$s (multi-head). Single-head networks increase their $\tau$ with $N$ and are able to solve tasks for large $N$, but they suffer from catastrophic forgetting. However, multi-head networks, which are explicitly required to hold multiple concurrent memories, keep $\tau$ constant and develop longer timescales through recurrent connectivity. Moreover, we show that the multi-head curriculum increases training speed and network stability to ablations and perturbations, and allows RNNs to generalize better to tasks beyond their training regime. This curriculum also significantly improves training GRUs and LSTMs for large-$N$ tasks. Our results suggest that adapting timescales to task requirements via recurrent interactions allows learning more complex objectives and improves the RNN's performance.

Tasks that require information about the world imply a trade-off between the time spent on observation and the variance of the response. In particular, fast decisions need to rely on uncertain information. However, standard estimates of information processing capabilities, such as the dynamic range, are defined based on mean values that assume infinite observation times. Here, we show that limiting the observation time results in distributions of responses whose variance increases with the temporal correlations in a system and, importantly, affects a system's confidence in distinguishing inputs and thereby making decisions. To quantify the ability to distinguish features of an input, we propose several measures and demonstrate them on the prime example of a recurrent neural network that represents an input rate by a response firing averaged over a finite observation time. We show analytically and in simulations that the optimal tuning of the network depends on the available observation time, implying that tasks require a ``useful'' rather than maximal sensitivity. Interestingly, this shifts the optimal dynamic regime from critical to subcritical for finite observation times and highlights the importance of incorporating the finite observation times concept in future studies of information processing capabilities in a principled manner., Comment: 6 pages, 3 figures; methods: 4 pages, 1 figure; supplementary. Comments and suggestions are welcome

Biological cortical neurons are remarkably sophisticated computational devices, temporally integrating their vast synaptic input over an intricate dendritic tree, subject to complex, nonlinearly interacting internal biological processes. A recent study proposed to characterize this complexity by fitting accurate surrogate models to replicate the input-output relationship of a detailed biophysical cortical pyramidal neuron model and discovered it needed temporal convolutional networks (TCN) with millions of parameters. Requiring these many parameters, however, could stem from a misalignment between the inductive biases of the TCN and cortical neuron's computations. In light of this, and to explore the computational implications of leaky memory units and nonlinear dendritic processing, we introduce the Expressive Leaky Memory (ELM) neuron model, a biologically inspired phenomenological model of a cortical neuron. Remarkably, by exploiting such slowly decaying memory-like hidden states and two-layered nonlinear integration of synaptic input, our ELM neuron can accurately match the aforementioned input-output relationship with under ten thousand trainable parameters. To further assess the computational ramifications of our neuron design, we evaluate it on various tasks with demanding temporal structures, including the Long Range Arena (LRA) datasets, as well as a novel neuromorphic dataset based on the Spiking Heidelberg Digits dataset (SHD-Adding). Leveraging a larger number of memory units with sufficiently long timescales, and correspondingly sophisticated synaptic integration, the ELM neuron displays substantial long-range processing capabilities, reliably outperforming the classic Transformer or Chrono-LSTM architectures on LRA, and even solving the Pathfinder-X task with over 70% accuracy (16k context length)., Comment: 25 pages, 14 figures, 13 tables, additional experiments and clarifications, accepted to ICLR 2024

The essential ingredient for studying the phenomena of emergence is the ability to generate and manipulate emergent systems that span large scales. Cellular automata are the model class particularly known for their effective scalability but are also typically constrained by fixed local rules. In this paper, we propose a new model class of adaptive cellular automata that allows for the generation of scalable and expressive models. We show how to implement computation-effective adaptation by coupling the update rule of the cellular automaton with itself and the system state in a localized way. To demonstrate the applications of this approach, we implement two different emergent models: a self-organizing Ising model and two types of plastic neural networks, a rate and spiking model. With the Ising model, we show how coupling local/global temperatures to local/global measurements can tune the model to stay in the vicinity of the critical temperature. With the neural models, we reproduce a classical balanced state in large recurrent neuronal networks with excitatory and inhibitory neurons and various plasticity mechanisms. Our study opens multiple directions for studying collective behavior and emergence.

It has long been hypothesized that operating close to the critical state is beneficial for natural, artificial and their evolutionary systems. We put this hypothesis to test in a system of evolving foraging agents controlled by neural networks that can adapt agents' dynamical regime throughout evolution. Surprisingly, we find that all populations that discover solutions, evolve to be subcritical. By a resilience analysis, we find that there are still benefits of starting the evolution in the critical regime. Namely, initially critical agents maintain their fitness level under environmental changes (for example, in the lifespan) and degrade gracefully when their genome is perturbed. At the same time, initially subcritical agents, even when evolved to the same fitness, are often inadequate to withstand the changes in the lifespan and degrade catastrophically with genetic perturbations. Furthermore, we find the optimal distance to criticality depends on the task complexity. To test it we introduce a hard and simple task: for the hard task, agents evolve closer to criticality whereas more subcritical solutions are found for the simple task. We verify that our results are independent of the selected evolutionary mechanisms by testing them on two principally different approaches: a genetic algorithm and an evolutionary strategy. In summary, our study suggests that although optimal behaviour in the simple task is obtained in a subcritical regime, initializing near criticality is important to be efficient at finding optimal solutions for new tasks of unknown complexity., Comment: arXiv admin note: substantial text overlap with arXiv:2103.12184

The evolutionary balance between innate and learned behaviors is highly intricate, and different organisms have found different solutions to this problem. We hypothesize that the emergence and exact form of learning behaviors is naturally connected with the statistics of environmental fluctuations and tasks an organism needs to solve. Here, we study how different aspects of simulated environments shape an evolved synaptic plasticity rule in static and moving artificial agents. We demonstrate that environmental fluctuation and uncertainty control the reliance of artificial organisms on plasticity. Interestingly, the form of the emerging plasticity rule is additionally determined by the details of the task the artificial organisms are aiming to solve. Moreover, we show that co-evolution between static connectivity and interacting plasticity mechanisms in distinct sub-networks changes the function and form of the emerging plasticity rules in embodied agents performing a foraging task.

Multiple studies of neural avalanches across different data modalities led to the prominent hypothesis that the brain operates near a critical point. The observed exponents often indicate the mean-field directed-percolation universality class, leading to the fully-connected or random network models to study the avalanche dynamics. However, the cortical networks have distinct non-random features and spatial organization that is known to affect the critical exponents. Here we show that distinct empirical exponents arise in networks with different topology and depend on the network size. In particular, we find apparent scale-free behavior with mean-field exponents appearing as quasi-critical dynamics in structured networks. This quasi-critical dynamics cannot be easily discriminated from an actual critical point in small networks. We find that the local coalescence in activity dynamics can explain the distinct exponents. Therefore, both topology and system size should be considered when assessing criticality from empirical observables.

Complex systems are fascinating because their rich macroscopic properties emerge from the interaction of many simple parts. Understanding the building principles of these emergent phenomena in nature requires assessing natural complex systems experimentally. However, despite the development of large-scale data-acquisition techniques, experimental observations are often limited to a tiny fraction of the system. This spatial subsampling is particularly severe in neuroscience, where only a tiny fraction of millions or even billions of neurons can be individually recorded. Spatial subsampling may lead to significant systematic biases when inferring the collective properties of the entire system naively from a subsampled part. To overcome such biases, powerful mathematical tools have been developed in the past. In this perspective, we overview some issues arising from subsampling and review recently developed approaches to tackle the subsampling problem. These approaches enable one to assess, e.g., graph structures, collective dynamics of animals, neural network activity, or the spread of disease correctly from observing only a tiny fraction of the system. However, our current approaches are still far from having solved the subsampling problem in general, and hence we conclude by outlining what we believe are the main open challenges. Solving these challenges alongside the development of large-scale recording techniques will enable further fundamental insights into the working of complex and living systems., Comment: 20 pages, 6 figures, review article

Correlated fluctuations in the activity of neural populations reflect the network's dynamics and connectivity. The temporal and spatial dimensions of neural correlations are interdependent. However, prior theoretical work mainly analyzed correlations in either spatial or temporal domains, oblivious to their interplay. We show that the network dynamics and connectivity jointly define the spatiotemporal profile of neural correlations. We derive analytical expressions for pairwise correlations in networks of binary units with spatially arranged connectivity in one and two dimensions. We find that spatial interactions among units generate multiple timescales in auto- and cross-correlations. Each timescale is associated with fluctuations at a particular spatial frequency, making a hierarchical contribution to the correlations. External inputs can modulate the correlation timescales when spatial interactions are nonlinear, and the modulation effect depends on the operating regime of network dynamics. These theoretical results open new ways to relate connectivity and dynamics in cortical networks via measurements of spatiotemporal neural correlations., Comment: 25 pages, 20 figures

Can we generate abstract aesthetic images without bias from natural or human selected image corpi? Are aesthetic images singled out in their correlation functions? In this paper we give answers to these and more questions. We generate images using compositional pattern-producing networks with random weights and varying architecture. We demonstrate that even with the randomly selected weights the correlation functions remain largely determined by the network architecture. In a controlled experiment, human subjects picked aesthetic images out of a large dataset of all generated images. Statistical analysis reveals that the correlation function is indeed different for aesthetic images.

Weights and directionality of the edges carry a large part of the information we can extract from a complex network. However, many network measures were formulated initially for undirected binary networks. The necessity to incorporate information about the weights led to the conception of the multiple extensions, particularly for definitions of the local clustering coefficient discussed here. We uncover that not all of these extensions are fully-weighted; some depend on the degree and thus change a lot when an infinitely small weight edge is exchanged for the absence of an edge, a feature that is not always desirable. We call these methods ``hybrid'' and argue that, in many situations, one should prefer fully-weighted definitions. After listing the necessary requirements for a method to analyze many various weighted networks properly, we propose a fully-weighted continuous clustering coefficient that satisfies all the previously proposed criteria while also being continuous with respect to vanishing weights. We demonstrate that the behavior and meaning of the Zhang--Horvath clustering and our new continuous definition provide complementary results and significantly outperform other definitions in multiple relevant conditions. Using synthetic and real-world examples, we show that when the network is inferred, noisy, or very heterogeneous, it is essential to use the fully-weighted clustering definitions., Comment: 18 pages, 8 figures, 10 tables (including appendix)

It has long been hypothesized that operating close to the critical state is beneficial for natural and artificial systems. We test this hypothesis by evolving foraging agents controlled by neural networks that can change the system's dynamical regime throughout evolution. Surprisingly, we find that all populations, regardless of their initial regime, evolve to be subcritical in simple tasks and even strongly subcritical populations can reach comparable performance. We hypothesize that the moderately subcritical regime combines the benefits of generalizability and adaptability brought by closeness to criticality with the stability of the dynamics characteristic for subcritical systems. By a resilience analysis, we find that initially critical agents maintain their fitness level even under environmental changes and degrade slowly with increasing perturbation strength. On the other hand, subcritical agents originally evolved to the same fitness, were often rendered utterly inadequate and degraded faster. We conclude that although the subcritical regime is preferable for a simple task, the optimal deviation from criticality depends on the task difficulty: for harder tasks, agents evolve closer to criticality. Furthermore, subcritical populations cannot find the path to decrease their distance to criticality. In summary, our study suggests that initializing models near criticality is important to find an optimal and flexible solution., Comment: 8 Pages, 7 Figures, Artificial Life Conference 2021

Self-organized criticality has been proposed to be a universal mechanism for the emergence of scale-free dynamics in many complex systems, and possibly in the brain. While such scale-free patterns were identified experimentally in many different types of neural recordings, the biological principles behind their emergence remained unknown. Utilizing different network models and motivated by experimental observations, synaptic plasticity was proposed as a possible mechanism to self-organize brain dynamics towards a critical point. In this review, we discuss how various biologically plausible plasticity rules operating across multiple timescales are implemented in the models and how they alter the network's dynamical state through modification of number and strength of the connections between the neurons. Some of these rules help to stabilize criticality, some need additional mechanisms to prevent divergence from the critical state. We propose that rules that are capable of bringing the network to criticality can be classified by how long the near-critical dynamics persists after their disabling. Finally, we discuss the role of self-organization and criticality in computation. Overall, the concept of criticality helps to shed light on brain function and self-organization, yet the overall dynamics of living neural networks seem to harnesses not only criticality for computation, but also deviations thereof., Comment: 34 pages, 7 figures

We propose a nonparametric parameter estimation of confidence intervals when the underlying has large or infinite variance. We explain the method by a simple numerical example and provide an application to estimate the coupling strength in neuronal networks.

Spreading processes are conventionally monitored on a macroscopic level by counting the number of incidences over time. The spreading process can then be modeled either on the microscopic level, assuming an underlying interaction network, or directly on the macroscopic level, assuming that microscopic contributions are negligible. The macroscopic characteristics of both descriptions are commonly assumed to be identical. In this work, we show that these characteristics of microscopic and macroscopic descriptions can be different due to coalescence, i.e., a node being activated at the same time by multiple sources. In particular, we consider a (microscopic) branching network (probabilistic cellular automaton) with annealed connectivity disorder, record the macroscopic activity, and then approximate this activity by a (macroscopic) branching process. In this framework, we analytically calculate the effect of coalescence on the collective dynamics. We show that coalescence leads to a universal non-linear scaling function for the conditional expectation value of successive network activity. This allows us to quantify the difference between the microscopic model parameter and established macroscopic estimates. To overcome this difference, we propose a non-linear estimator that correctly infers the model branching parameter for all system sizes., Comment: 13 pages

The dynamic range of stimulus processing in living organisms is much larger than a single neural network can explain. For a generic, tunable spiking network we derive that while the dynamic range is maximal at criticality, the interval of discriminable intensities is very similar for any network tuning due to coalescence. Compensating coalescence enables adaptation of discriminable intervals. Thus, we can tailor an ensemble of networks optimized to the distribution of stimulus intensities, e.g., extending the dynamic range arbitrarily. We discuss potential applications in machine learning., Comment: 6 pages plus supplemental material

Power laws in nature are considered to be signatures of complexity. The theory of self-organized criticality (SOC) was proposed to explain their origins. A longstanding principle of SOC is the \emph{separation of timescales} axiom. It dictates that external input is delivered to the system at a much slower rate compared to the timescale of internal dynamics. The statistics of neural avalanches in the brain was demonstrated to follow a power law, indicating closeness to a critical state. Moreover, criticality was shown to be a beneficial state for various computations leading to the hypothesis, that the brain is a SOC system. However, for neuronal systems that are constantly bombarded by incoming signals, separation of timescales assumption is unnatural. Recently it was experimentally demonstrated that a proper correction of the avalanche detection algorithm to account for the increased drive during task performance leads to a change of the power-law exponent from $1.5$ to approximately $1.3$, but there is so far no theoretical explanation for this change. Here we investigate the importance of timescales separation, by partly abandoning it in various models. We achieve it by allowing for external input during the avalanche, without compromising the separation of avalanches. We develop an analytic treatment and provide numerical simulations of a simple neuronal model.

We define the Abelian distribution and study its basic properties. Abelian distributions arise in the context of neural modeling and describe the size of neural avalanches in fully-connected integrate-and-fire models of self-organized criticality in neural systems.

In real-world applications, observations are often constrained to a small fraction of a system. Such spatial subsampling can be caused by the inaccessibility or the sheer size of the system, and cannot be overcome by longer sampling. Spatial subsampling can strongly bias inferences about a system's aggregated properties. To overcome the bias, we derive analytically a subsampling scaling framework that is applicable to different observables, including distributions of neuronal avalanches, of number of people infected during an epidemic outbreak, and of node degrees. We demonstrate how to infer the correct distributions of the underlying full system, how to apply it to distinguish critical from subcritical systems, and how to disentangle subsampling and finite size effects. Lastly, we apply subsampling scaling to neuronal avalanche models and to recordings from developing neural networks. We show that only mature, but not young networks follow power-law scaling, indicating self-organization to criticality during development., Comment: 33 pages, including 15 pages of supplementary

How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness upon gradual addition of links, was long believed to be continuous but recent numerical evidence on "explosive percolation" suggests that it might as well be discontinuous if links compete for addition. Here we analyze the microscopic mechanisms underlying discontinuous percolation processes and reveal a strong impact of single link additions. We show that in generic competitive percolation processes, including those displaying explosive percolation, single links do not induce a discontinuous gap in the largest cluster size in the thermodynamic limit. Nevertheless, our results highlight that for large finite systems single links may still induce observable gaps because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, thus announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively., Comment: non-final version, for final see Nature Physics homepage

We show that a network of spiking neurons exhibits robust self-organized criticality if the synaptic efficacies follow realistic dynamics. Deriving analytical expressions for the average coupling strengths and inter-spike intervals, we demonstrate that networks with dynamical synapses exhibit critical avalanche dynamics for a wide range of interaction parameters. We prove that in the thermodynamical limit the network becomes critical for all large enough coupling parameters. We thereby explain experimental observations in which cortical neurons show avalanche activity with the total intensity of firing events being distributed as a power-law., Comment: 9 pages, 4 figures