John Beggs

Networks of living neurons exhibit diverse patterns of activity, including oscillations, synchrony, and waves. Recent work in physics has shown yet another mode of activity in systems composed of many nonlinear units interacting locally. For example, avalanches, earthquakes, and forest fires all propagate in systems organized into a critical state in which event sizes show no characteristic scale and are described by power laws. We hypothesized that a similar mode of activity with complex emergent properties could exist in networks of cortical neurons. We investigated this issue in mature organotypic cultures and acute slices of rat cortex by recording spontaneous local field potentials continuously using a 60 channel multielectrode array. Here, we show that propagation of spontaneous activity in cortical networks is described by equations that govern avalanches. As predicted by theory for a critical branching process …

A major goal of neuroscience is to elucidate mechanisms of cortical information processing and storage. Previous work from our laboratory (Beggs and Plenz, 2003) revealed that propagation of local field potentials (LFPs) in cortical circuits could be described by the same equations that govern avalanches. Whereas modeling studies suggested that these “neuronal avalanches” were optimal for information transmission, it was not clear what role they could play in information storage. Work from numerous other laboratories has shown that cortical structures can generate reproducible spatiotemporal patterns of activity that could be used as a substrate for memory. Here, we show that although neuronal avalanches lasted only a few milliseconds, their spatiotemporal patterns were also stable and significantly repeatable even many hours later. To investigate these issues, we cultured coronal slices of rat cortex for 4 …

Recent experimental work has shown that activity in living neural networks can propagate as a critical branching process that revisits many metastable states. Neural network theory suggests that attracting states could store information, but little is known about how a branching process could form such states. Here we use a branching process to model actual data and to explore metastable states in the network. When we tune the branching parameter to the critical point, we find that metastable states are most numerous and that network dynamics are not attracting, but neutral.

Early theoretical and simulation work independently undertaken by Packard, Langton and Kauffman suggested that adaptability and computational power would be optimized in systems at the ‘edge of chaos’, at a critical point in a phase transition between total randomness and boring order. This provocative hypothesis has received much attention, but biological experiments supporting it have been relatively few. Here, we review recent experiments on networks of cortical neurons, showing that they appear to be operating near the critical point. Simulation studies capture the main features of these data and suggest that criticality may allow cortical networks to optimize information processing. These simulations lead to predictions that could be tested in the near future, possibly providing further experimental evidence for the criticality hypothesis.

Relatively recent work has reported that networks of neurons can produce avalanches of activity whose sizes follow a power law distribution. This suggests that these networks may be operating near a critical point, poised between a phase where activity rapidly dies out and a phase where activity is amplified over time. The hypothesis that the electrical activity of neural networks in the brain is critical is potentially important, as many simulations suggest that information processing functions would be optimized at the critical point. This hypothesis, however, is still controversial. Here we will explain the concept of criticality and review the substantial objections to the criticality hypothesis raised by skeptics. Points and counter points are presented in dialogue form.

Multineuron firing patterns are often observed, yet are predicted to be rare by models that assume independent firing. To explain these correlated network states, two groups recently applied a second-order maximum entropy model that used only observed firing rates and pairwise interactions as parameters (Schneidman et al., 2006; Shlens et al., 2006). Interestingly, with these minimal assumptions they predicted 90–99% of network correlations. If generally applicable, this approach could vastly simplify analyses of complex networks. However, this initial work was done largely on retinal tissue, and its applicability to cortical circuits is mostly unknown. This work also did not address the temporal evolution of correlated states. To investigate these issues, we applied the model to multielectrode data containing spontaneous spikes or local field potentials from cortical slices and cultures. The model worked slightly less …

The tasks of neural computation are remarkably diverse. To function optimally, neuronal networks have been hypothesized to operate near a nonequilibrium critical point. However, experimental evidence for critical dynamics has been inconclusive. Here, we show that the dynamics of cultured cortical networks are critical. We analyze neuronal network data collected at the individual neuron level using the framework of nonequilibrium phase transitions. Among the most striking predictions confirmed is that the mean temporal profiles of avalanches of widely varying durations are quantitatively described by a single universal scaling function. We also show that the data have three additional features predicted by critical phenomena: approximate power law distributions of avalanche sizes and durations, samples in subcritical and supercritical phases, and scaling laws between anomalous exponents.

Transfer entropy (TE) is an information-theoretic measure which has received recent attention in neuroscience for its potential to identify effective connectivity between neurons. Calculating TE for large ensembles of spiking neurons is computationally intensive, and has caused most investigators to probe neural interactions at only a single time delay and at a message length of only a single time bin. This is problematic, as synaptic delays between cortical neurons, for example, range from one to tens of milliseconds. In addition, neurons produce bursts of spikes spanning multiple time bins. To address these issues, here we introduce a free software package that allows TE to be measured at multiple delays and message lengths. To assess performance, we applied these extensions of TE to a spiking cortical network model (Izhikevich, 2006) with known connectivity and a range of synaptic delays. For comparison, we also investigated single-delay TE, at a message length of one bin (D1TE), and cross-correlation (CC) methods. We found that D1TE could identify 36% of true connections when evaluated at a false positive rate of 1%. For extended versions of TE, this dramatically improved to 73% of true connections. In addition, the connections correctly identified by extended versions of TE accounted for 85% of the total synaptic weight in the network. Cross correlation methods generally performed more poorly than extended TE, but were useful when data length was short. A computational performance analysis demonstrated that the algorithm for extended TE, when used on currently available desktop computers, could extract effective connectivity …

Information theory has long been used to quantify interactions between two variables. With the rise of complex systems research, multivariate information measures have been increasingly used to investigate interactions between groups of three or more variables, often with an emphasis on so called synergistic and redundant interactions. While bivariate information measures are commonly agreed upon, the multivariate information measures in use today have been developed by many different groups, and differ in subtle, yet significant ways. Here, we will review these multivariate information measures with special emphasis paid to their relationship to synergy and redundancy, as well as examine the differences between these measures by applying them to several simple model systems. In addition to these systems, we will illustrate the usefulness of the information measures by analyzing neural spiking …

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 …

The performance of complex networks, like the brain, depends on how effectively their elements communicate. Despite the importance of communication, it is virtually unknown how information is transferred in local cortical networks, consisting of hundreds of closely spaced neurons. To address this, it is important to record simultaneously from hundreds of neurons at a spacing that matches typical axonal connection distances, and at a temporal resolution that matches synaptic delays. We used a 512-electrode array (60 μm spacing) to record spontaneous activity at 20 kHz from up to 500 neurons simultaneously in slice cultures of mouse somatosensory cortex for 1 h at a time. We applied a previously validated version of transfer entropy to quantify information transfer. Similar to in vivo reports, we found an approximately lognormal distribution of firing rates. Pairwise information transfer strengths also were nearly …

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Layer II/III of rat perirhinal cortex (PR) contains numerous late-spiking (LS) pyramidal neurons. When injected with a depolarizing current step, these LS cells typically delay spiking for one or more seconds from the onset of the current step and thensustain firing for the duration of the step. This pattern of delayed and sustained firing suggested a specific computational role for LS cells in temporal learning. This hypothesis predicts and requires that some layer II/III neurons should also exhibit delayed and sustained spiking in response to a train of excitatorysynaptic inputs. Here we tested this prediction using visually guided, whole cell recordings from rat PR brain slices. Most LS cells (19 of 26) exhibited delayed spiking to synaptic stimulation (>1 s latency from the train onset), and the majority of these cells (13 of 19) also showed sustained firing that persisted for the duration of the synaptic train (5–10 s duration …

Is the brain really operating at a critical point? We study the nonequilibrium properties of a neural network which models the dynamics of the neocortex and argue for optimal quasicritical dynamics on the Widom line where the correlation length and information transmission are optimized. We simulate the network and introduce an analytical mean-field approximation, characterize the nonequilibrium phase transitions, and present a nonequilibrium phase diagram, which shows that in addition to an ordered and disordered phase, the system exhibits a “quasiperiodic” phase corresponding to synchronous activity in simulations, which may be related to the pathological synchronization associated with epilepsy.

The dynamics of microelectrode local field potentials from cortical slice cultures shows critical behavior. A desirable feature of criticality is that information transmission is optimal in this state. We explore a biologically plausible neural net model that can dynamically converge on criticality and that can return to criticality if perturbed away from it. Our model assumes the presence of a preferred target firing rate, with dynamical adjustments of internodal connection strengths to approach this firing rate. We suggest that mechanisms for maintaining firing rate homeostasis may also maintain a neural system at criticality.