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1.  Multiplex Networks of Cortical and Hippocampal Neurons Revealed at Different Timescales 
PLoS ONE  2014;9(12):e115764.
Recent studies have emphasized the importance of multiplex networks – interdependent networks with shared nodes and different types of connections – in systems primarily outside of neuroscience. Though the multiplex properties of networks are frequently not considered, most networks are actually multiplex networks and the multiplex specific features of networks can greatly affect network behavior (e.g. fault tolerance). Thus, the study of networks of neurons could potentially be greatly enhanced using a multiplex perspective. Given the wide range of temporally dependent rhythms and phenomena present in neural systems, we chose to examine multiplex networks of individual neurons with time scale dependent connections. To study these networks, we used transfer entropy – an information theoretic quantity that can be used to measure linear and nonlinear interactions – to systematically measure the connectivity between individual neurons at different time scales in cortical and hippocampal slice cultures. We recorded the spiking activity of almost 12,000 neurons across 60 tissue samples using a 512-electrode array with 60 micrometer inter-electrode spacing and 50 microsecond temporal resolution. To the best of our knowledge, this preparation and recording method represents a superior combination of number of recorded neurons and temporal and spatial recording resolutions to any currently available in vivo system. We found that highly connected neurons (“hubs”) were localized to certain time scales, which, we hypothesize, increases the fault tolerance of the network. Conversely, a large proportion of non-hub neurons were not localized to certain time scales. In addition, we found that long and short time scale connectivity was uncorrelated. Finally, we found that long time scale networks were significantly less modular and more disassortative than short time scale networks in both tissue types. As far as we are aware, this analysis represents the first systematic study of temporally dependent multiplex networks among individual neurons.
PMCID: PMC4275261  PMID: 25536059
2.  Large-Scale, High-Resolution Multielectrode-Array Recording Depicts Functional Network Differences of Cortical and Hippocampal Cultures 
PLoS ONE  2014;9(8):e105324.
Understanding the detailed circuitry of functioning neuronal networks is one of the major goals of neuroscience. Recent improvements in neuronal recording techniques have made it possible to record the spiking activity from hundreds of neurons simultaneously with sub-millisecond temporal resolution. Here we used a 512-channel multielectrode array system to record the activity from hundreds of neurons in organotypic cultures of cortico-hippocampal brain slices from mice. To probe the network structure, we employed a wavelet transform of the cross-correlogram to categorize the functional connectivity in different frequency ranges. With this method we directly compare, for the first time, in any preparation, the neuronal network structures of cortex and hippocampus, on the scale of hundreds of neurons, with sub-millisecond time resolution. Among the three frequency ranges that we investigated, the lower two frequency ranges (gamma (30–80 Hz) and beta (12–30 Hz) range) showed similar network structure between cortex and hippocampus, but there were many significant differences between these structures in the high frequency range (100–1000 Hz). The high frequency networks in cortex showed short tailed degree-distributions, shorter decay length of connectivity density, smaller clustering coefficients, and positive assortativity. Our results suggest that our method can characterize frequency dependent differences of network architecture from different brain regions. Crucially, because these differences between brain regions require millisecond temporal scales to be observed and characterized, these results underscore the importance of high temporal resolution recordings for the understanding of functional networks in neuronal systems.
PMCID: PMC4134292  PMID: 25126851
3.  Network community, clusters and hubs in cortical micro circuits 
BMC Neuroscience  2014;15(Suppl 1):F2.
PMCID: PMC4124982
5.  Patterns of information flow in local cortical networks 
BMC Neuroscience  2014;15(Suppl 1):P213.
PMCID: PMC4126424
6.  Behavior Modulates Effective Connectivity between Cortex and Striatum 
PLoS ONE  2014;9(3):e89443.
It has been notoriously difficult to understand interactions in the basal ganglia because of multiple recurrent loops. Another complication is that activity there is strongly dependent on behavior, suggesting that directional interactions, or effective connections, can dynamically change. A simplifying approach would be to examine just the direct, monosynaptic projections from cortex to striatum and contrast this with the polysynaptic feedback connections from striatum to cortex. Previous work by others on effective connectivity in this pathway indicated that activity in cortex could be used to predict activity in striatum, but that striatal activity could not predict cortical activity. However, this work was conducted in anesthetized or seizing animals, making it impossible to know how free behavior might influence effective connectivity. To address this issue, we applied Granger causality to local field potential signals from cortex and striatum in freely behaving rats. Consistent with previous results, we found that effective connectivity was largely unidirectional, from cortex to striatum, during anesthetized and resting states. Interestingly, we found that effective connectivity became bidirectional during free behaviors. These results are the first to our knowledge to show that striatal influence on cortex can be as strong as cortical influence on striatum. In addition, these findings highlight how behavioral states can affect basal ganglia interactions. Finally, we suggest that this approach may be useful for studies of Parkinson's or Huntington's diseases, in which effective connectivity may change during movement.
PMCID: PMC3949668  PMID: 24618981
7.  Being Critical of Criticality in the Brain 
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 dialog form.
PMCID: PMC3369250  PMID: 22701101
criticality; scale-free; avalanche; network; multi-electrode array; statistical physics; Ising model
8.  Extending Transfer Entropy Improves Identification of Effective Connectivity in a Spiking Cortical Network Model 
PLoS ONE  2011;6(11):e27431.
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 from 1 hr recordings containing 200 neurons in ∼5 min. We conclude that extending TE to multiple delays and message lengths improves its ability to assess effective connectivity between spiking neurons. These extensions to TE soon could become practical tools for experimentalists who record hundreds of spiking neurons.
PMCID: PMC3216957  PMID: 22102894
9.  A few strong connections: optimizing information retention in neuronal avalanches 
BMC Neuroscience  2010;11:3.
How living neural networks retain information is still incompletely understood. Two prominent ideas on this topic have developed in parallel, but have remained somewhat unconnected. The first of these, the "synaptic hypothesis," holds that information can be retained in synaptic connection strengths, or weights, between neurons. Recent work inspired by statistical mechanics has suggested that networks will retain the most information when their weights are distributed in a skewed manner, with many weak weights and only a few strong ones. The second of these ideas is that information can be represented by stable activity patterns. Multineuron recordings have shown that sequences of neural activity distributed over many neurons are repeated above chance levels when animals perform well-learned tasks. Although these two ideas are compelling, no one to our knowledge has yet linked the predicted optimum distribution of weights to stable activity patterns actually observed in living neural networks.
Here, we explore this link by comparing stable activity patterns from cortical slice networks recorded with multielectrode arrays to stable patterns produced by a model with a tunable weight distribution. This model was previously shown to capture central features of the dynamics in these slice networks, including neuronal avalanche cascades. We find that when the model weight distribution is appropriately skewed, it correctly matches the distribution of repeating patterns observed in the data. In addition, this same distribution of weights maximizes the capacity of the network model to retain stable activity patterns. Thus, the distribution that best fits the data is also the distribution that maximizes the number of stable patterns.
We conclude that local cortical networks are very likely to use a highly skewed weight distribution to optimize information retention, as predicted by theory. Fixed distributions impose constraints on learning, however. The network must have mechanisms for preserving the overall weight distribution while allowing individual connection strengths to change with learning.
PMCID: PMC2824798  PMID: 20053290
10.  An open hypothesis: is epilepsy learned, and can it be unlearned? 
Epilepsy & behavior : E&B  2008;13(3):511-522.
Plasticity is central to the ability of a neural system to learn and also to its ability to develop spontaneous seizures. What is the connection between the two? Learning itself is known to be a destabilizing process at the algorithmic level. We have investigated necessary constraints on a spontaneously active Hebbian learning system and find that the ability to learn appears to confer an intrinsic vulnerability to epileptogenesis on that system. We hypothesize that epilepsy arises as an abnormal learned response of such a system to certain repeated provocations. This response is a network level effect. If epilepsy really is a learned response, then it should be possible to reverse it, i.e., to unlearn epilepsy. Unlearning epilepsy may then provide a new approach to its treatment.
PMCID: PMC2611958  PMID: 18573694
epileptogenesis; critical connectivity; homeostasis; Hebbian learning

Results 1-10 (10)