Transport of information along unidimensional layered networks of dissociated hippocampal neurons and implications for rate coding

Microfluidics for Neuronal Cell and Circuit Engineering

The widespread adoption of microfluidic devices among the neuroscience and neurobiology communities has enabled addressing a broad range of questions at the molecular, cellular, circuit, and system levels. Here, we review biomedical engineering approaches that harness the power of microfluidics for bottom-up generation of neuronal cell types and for the assembly and analysis of neural circuits. Microfluidics-based approaches are instrumental to generate the knowledge necessary for the derivation of diverse neuronal cell types from human pluripotent stem cells, as they enable the isolation and subsequent examination of individual neurons of interest. Moreover, microfluidic devices allow to engineer neural circuits with specific orientations and directionality by providing control over neuronal cell polarity and permitting the isolation of axons in individual microchannels. Similarly, the use of microfluidic chips enables the construction not only of 2D but also of 3D brain, retinal, and peripheral nervous system model circuits. Such brain-on-a-chip and organoid-on-a-chip technologies are promising platforms for studying these organs as they closely recapitulate some aspects of in vivo biological processes. Microfluidic 3D neuronal models, together with 2D in vitro systems, are widely used in many applications ranging from drug development and toxicology studies to neurological disease modeling and personalized medicine. Altogether, microfluidics provide researchers with powerful systems that complement and partially replace animal models.

Propagation of temporal and rate signals in cultured multilayer networks

Analyses of idealized feedforward networks suggest that several conditions have to be satisfied in order for activity to propagate faithfully across layers. Verifying these concepts experimentally has been difficult owing to the vast number of variables that must be controlled. Here, we cultured cortical neurons in a chamber with sequentially connected compartments, optogenetically stimulated individual neurons in the first layer with high spatiotemporal resolution, and then monitored the subthreshold and suprathreshold potentials in subsequent layers. Brief stimuli delivered to the first layer evoked a short-latency transient response followed by sustained activity. Rate signals, carried by the sustained component, propagated reliably through 4 layers, unlike idealized feedforward networks, which tended strongly towards synchrony. Moreover, temporal jitter in the stimulus was transformed into a rate code and transmitted to the last layer. This novel mode of propagation occurred in the balanced excitatory-inhibitory regime and is mediated by NMDA-mediated receptors and recurrent activity.

Analog control with two Artificial Axons

The artificial axon is a recently introduced synthetic assembly of supported lipid bilayers and voltage gated ion channels, displaying the basic electrophysiology of nerve cells. Here we demonstrate the use of two artificial axons as control elements to achieve a simple task. Namely, we steer a remote control car towards a light source, using the sensory input dependent firing rate of the axons as the control signal for turning left or right. We present the result in the form of the analysis of a movie of the car approaching the light source. In general terms, with this work we pursue a constructivist approach to exploring the nexus between machine language at the nerve cell level and behavior.

In vitro neural networks minimise variational free energy

In this work, we address the neuronal encoding problem from a Bayesian perspective. Specifically, we ask whether neuronal responses in an in vitro neuronal network are consistent with ideal Bayesian observer responses under the free energy principle. In brief, we stimulated an in vitro cortical cell culture with stimulus trains that had a known statistical structure. We then asked whether recorded neuronal responses were consistent with variational message passing based upon free energy minimisation (i.e., evidence maximisation). Effectively, this required us to solve two problems: first, we had to formulate the Bayes-optimal encoding of the causes or sources of sensory stimulation, and then show that these idealised responses could account for observed electrophysiological responses. We describe a simulation of an optimal neural network (i.e., the ideal Bayesian neural code) and then consider the mapping from idealised in silico responses to recorded in vitro responses. Our objective was to find evidence for functional specialisation and segregation in the in vitro neural network that reproduced in silico learning via free energy minimisation. Finally, we combined the in vitro and in silico results to characterise learning in terms of trajectories in a variational information plane of accuracy and complexity.

Modeling the neuron as a nanocommunication system to identify spatiotemporal molecular events in neurodegenerative disease

AIM

In tauopathies such as Alzheimer's disease (AD), molecular changes spanning multiple subcellular compartments of the neuron contribute to neurodegeneration and altered axonal signaling. Computational modeling of end-to-end linked events benefit mechanistic analysis and can be informative to understand disease progression and accelerate development of effective therapies. In the calcium-amyloid beta model of AD, calcium ions that are an important regulator of neuronal function undergo dysregulated homeostasis that disrupts cargo loading for neurotrophic signaling along axonal microtubules (MTs). The aim of the present study was to develop a computational model of the neuron using a layered architecture simulation that enables us to evaluate the functionalities of several interlinked components in the calcium-amyloid beta model.

METHODS

The elevation of intracellular calcium levels is modeled upon binding of amyloid beta oligomers (AβOs) to calcium channels or as a result of membrane insertion of oligomeric Aβ1-42 to form pores/channels. The resulting subsequent Ca2+ disruption of dense core vesicle (DCV)-kinesin cargo loading and transport of brain-derived neurotrophic factor (BDNF) on axonal MTs are then evaluated. Our model applies published experimental data on calcium channel manipulation of DCV-BDNF and incorporates organizational complexity of the axon as bundled polar and discontinuous MTs. The interoperability simulation is based on the Institute of Electrical and Electronics Engineers standard association P1906.1 framework for nanoscale and molecular communication.

RESULTS

Our analysis provides new spatiotemporal insights into the end-to-end signaling events linking calcium dysregulation and BDNF transport and by simulation compares the relative impact of dysregulation of calcium levels by AβO-channel interactions, oligomeric Aβ1-42 pores/channel formation, and release of calcium by internal stores. The flexible platform of our model allows continued expansion of molecular details including mechanistic and morphological parameters of axonal cytoskeleton networks as they become available to test disease and treatment predictions.

CONCLUSION

The present model will benefit future drug studies on calcium homeostasis and dysregulation linked to measurable neural functional outcomes. The algorithms used can also link to other multiscale multi-cellular modeling platforms to fill in molecular gaps that we believe will assist in broadening and refining multiscale computational maps of neurodegeneration.

External Excitation of Neurons Using Electric and Magnetic Fields in One- and Two-dimensional Cultures

A neuron will fire an action potential when its membrane potential exceeds a certain threshold. In typical activity of the brain, this occurs as a result of chemical inputs to its synapses. However, neurons can also be excited by an imposed electric field. In particular, recent clinical applications activate neurons by creating an electric field externally. It is therefore of interest to investigate how the neuron responds to the external field and what causes the action potential. Fortunately, precise and controlled application of an external electric field is possible for embryonic neuronal cells that are excised, dissociated and grown in cultures. This allows the investigation of these questions in a highly reproducible system. In this paper some of the techniques used for controlled application of external electric field on neuronal cultures are reviewed. The networks can be either one dimensional, i.e. patterned in linear forms or allowed to grow on the whole plane of the substrate, and thus two dimensional. Furthermore, the excitation can be created by the direct application of electric field via electrodes immersed in the fluid (bath electrodes) or by inducing the electric field using the remote creation of magnetic pulses.

Structure, Function, and Propagation of Information across Living Two, Four, and Eight Node Degree Topologies

In this study, we created four network topologies composed of living cortical neurons and compared resultant structural-functional dynamics including the nature and quality of information transmission. Each living network was composed of living cortical neurons and were created using microstamping of adhesion promoting molecules and each was "designed" with different levels of convergence embedded within each structure. Networks were cultured over a grid of electrodes that permitted detailed measurements of neural activity at each node in the network. Of the topologies we tested, the "Random" networks in which neurons connect based on their own intrinsic properties transmitted information embedded within their spike trains with higher fidelity relative to any other topology we tested. Within our patterned topologies in which we explicitly manipulated structure, the effect of convergence on fidelity was dependent on both topology and time-scale (rate vs. temporal coding). A more detailed examination using tools from network analysis revealed that these changes in fidelity were also associated with a number of other structural properties including a node's degree, degree-degree correlations, path length, and clustering coefficients. Whereas information transmission was apparent among nodes with few connections, the greatest transmission fidelity was achieved among the few nodes possessing the highest number of connections (high degree nodes or putative hubs). These results provide a unique view into the relationship between structure and its affect on transmission fidelity, at least within these small neural populations with defined network topology. They also highlight the potential role of tools such as microstamp printing and microelectrode array recordings to construct and record from arbitrary network topologies to provide a new direction in which to advance the study of structure-function relationships.

Mobile Microplates for Morphological Control and Assembly of Individual Neural Cells

A microfabricated device that enables morphological control and assembly of cultured single neural cells is described. Assembly of morphologically controlled single neural cells allows neuroengineers to design in vitro neural circuits with a single-cell resolution. Compared to conventional cell-patterning techniques, the device allows for the highly precise positioning of neural somas and neurites in a reproducible fashion.

Cultured Cortical Neurons Can Perform Blind Source Separation According to the Free-Energy Principle

Blind source separation is the computation underlying the cocktail party effect--a partygoer can distinguish a particular talker's voice from the ambient noise. Early studies indicated that the brain might use blind source separation as a signal processing strategy for sensory perception and numerous mathematical models have been proposed; however, it remains unclear how the neural networks extract particular sources from a complex mixture of inputs. We discovered that neurons in cultures of dissociated rat cortical cells could learn to represent particular sources while filtering out other signals. Specifically, the distinct classes of neurons in the culture learned to respond to the distinct sources after repeating training stimulation. Moreover, the neural network structures changed to reduce free energy, as predicted by the free-energy principle, a candidate unified theory of learning and memory, and by Jaynes' principle of maximum entropy. This implicit learning can only be explained by some form of Hebbian plasticity. These results are the first in vitro (as opposed to in silico) demonstration of neural networks performing blind source separation, and the first formal demonstration of neuronal self-organization under the free energy principle.

A mechanism for graded, dynamically routable current propagation in pulse-gated synfire chains and implications for information coding

Neural oscillations can enhance feature recognition (Azouz and Gray Proceedings of the National Academy of Sciences of the United States of America, 97, 8110-8115 2000), modulate interactions between neurons (Womelsdorf et al. Science, 316, 1609-01612 2007), and improve learning and memory (Markowska et al. The Journal of Neuroscience, 15, 2063-2073 1995). Numerical studies have shown that coherent spiking can give rise to windows in time during which information transfer can be enhanced in neuronal networks (Abeles Israel Journal of Medical Sciences, 18, 83-92 1982; Lisman and Idiart Science, 267, 1512-1515 1995, Salinas and Sejnowski Nature Reviews. Neuroscience, 2, 539-550 2001). Unanswered questions are: 1) What is the transfer mechanism? And 2) how well can a transfer be executed? Here, we present a pulse-based mechanism by which a graded current amplitude may be exactly propagated from one neuronal population to another. The mechanism relies on the downstream gating of mean synaptic current amplitude from one population of neurons to another via a pulse. Because transfer is pulse-based, information may be dynamically routed through a neural circuit with fixed connectivity. We demonstrate the transfer mechanism in a realistic network of spiking neurons and show that it is robust to noise in the form of pulse timing inaccuracies, random synaptic strengths and finite size effects. We also show that the mechanism is structurally robust in that it may be implemented using biologically realistic pulses. The transfer mechanism may be used as a building block for fast, complex information processing in neural circuits. We show that the mechanism naturally leads to a framework wherein neural information coding and processing can be considered as a product of linear maps under the active control of a pulse generator. Distinct control and processing components combine to form the basis for the binding, propagation, and processing of dynamically routed information within neural pathways. Using our framework, we construct example neural circuits to 1) maintain a short-term memory, 2) compute time-windowed Fourier transforms, and 3) perform spatial rotations. We postulate that such circuits, with automatic and stereotyped control and processing of information, are the neural correlates of Crick and Koch's zombie modes.

Oscillation-induced signal transmission and gating in neural circuits

Reliable signal transmission constitutes a key requirement for neural circuit function. The propagation of synchronous pulse packets through recurrent circuits is hypothesized to be one robust form of signal transmission and has been extensively studied in computational and theoretical works. Yet, although external or internally generated oscillations are ubiquitous across neural systems, their influence on such signal propagation is unclear. Here we systematically investigate the impact of oscillations on propagating synchrony. We find that for standard, additive couplings and a net excitatory effect of oscillations, robust propagation of synchrony is enabled in less prominent feed-forward structures than in systems without oscillations. In the presence of non-additive coupling (as mediated by fast dendritic spikes), even balanced oscillatory inputs may enable robust propagation. Here, emerging resonances create complex locking patterns between oscillations and spike synchrony. Interestingly, these resonances make the circuits capable of selecting specific pathways for signal transmission. Oscillations may thus promote reliable transmission and, in co-action with dendritic nonlinearities, provide a mechanism for information processing by selectively gating and routing of signals. Our results are of particular interest for the interpretation of sharp wave/ripple complexes in the hippocampus, where previously learned spike patterns are replayed in conjunction with global high-frequency oscillations. We suggest that the oscillations may serve to stabilize the replay.

Rhythmic activity in the brain is ubiquitous, its functions are debated. Here we show that it may contribute to the reliable transmission of information within brain areas. We find that its effect is particularly strong if we take nonlinear coupling into account. This experimentally found neuronal property implies that inputs which arrive nearly simultaneously can have a much stronger impact than expected from the sum of their individuals strengths. In such systems, rhythmic activity supports information transmission even if its positive and negative part exactly cancels all the time. Further, the information transmission can adapt to the oscillation frequency to optimally benefit from it. Finally, we show that rhythms with different frequencies may enable or disable communication channels, and are thus suitable for the steering of information flow.

Propagating synchrony in feed-forward networks

Coordinated patterns of precisely timed action potentials (spikes) emerge in a variety of neural circuits but their dynamical origin is still not well understood. One hypothesis states that synchronous activity propagating through feed-forward chains of groups of neurons (synfire chains) may dynamically generate such spike patterns. Additionally, synfire chains offer the possibility to enable reliable signal transmission. So far, mostly densely connected chains, often with all-to-all connectivity between groups, have been theoretically and computationally studied. Yet, such prominent feed-forward structures have not been observed experimentally. Here we analytically and numerically investigate under which conditions diluted feed-forward chains may exhibit synchrony propagation. In addition to conventional linear input summation, we study the impact of non-linear, non-additive summation accounting for the effect of fast dendritic spikes. The non-linearities promote synchronous inputs to generate precisely timed spikes. We identify how non-additive coupling relaxes the conditions on connectivity such that it enables synchrony propagation at connectivities substantially lower than required for linearly coupled chains. Although the analytical treatment is based on a simple leaky integrate-and-fire neuron model, we show how to generalize our methods to biologically more detailed neuron models and verify our results by numerical simulations with, e.g., Hodgkin Huxley type neurons.

Ultra-rapid laser protein micropatterning: screening for directed polarization of single neurons

Protein micropatterning is a powerful tool for studying the effects of extracellular signals on cell development and regeneration. Laser micropatterning of proteins is the most flexible method for patterning many different geometries, protein densities, and concentration gradients. Despite these advantages, laser micropatterning remains prohibitively slow for most applications. Here, we take advantage of the rapid multi-photon induced photobleaching of fluorophores to generate sub-micron resolution patterns of full-length proteins on polymer monolayers, with sub-microsecond exposure times, i.e. one to five orders of magnitude faster than all previous laser micropatterning methods. We screened a range of different PEG monolayer coupling chemistries, chain-lengths and functional caps, and found that long-chain acrylated PEG monolayers are effective at resisting non-specific protein adhesion, while permitting efficient cross-linking of biotin-4-fluorescein to the PEG monolayers upon exposure to femtosecond laser pulses. We find evidence that the dominant photopatterning chemistry switches from a two-photon process to three- and four-photon absorption processes as the laser intensity increases, generating increasingly volatile excited triplet-state fluorophores, leading to faster patterning. Using this technology, we were able to generate over a hundred thousand protein patterns with varying geometries and protein densities to direct the polarization of hippocampal neurons with single-cell precision. We found that certain arrays of patterned triangles as small as neurite growth cones can direct polarization by impeding the elongation of reverse-projecting neurites, while permitting elongation of forward-projecting neurites. The ability to rapidly generate and screen such protein micropatterns can enable discovery of conditions necessary to create in vitro neural networks with single-neuron precision for basic discovery, drug screening, as well as for tissue scaffolding in therapeutics.

Non-additive coupling enables propagation of synchronous spiking activity in purely random networks

Despite the current debate about the computational role of experimentally observed precise spike patterns it is still theoretically unclear under which conditions and how they may emerge in neural circuits. Here, we study spiking neural networks with non-additive dendritic interactions that were recently uncovered in single-neuron experiments. We show that supra-additive dendritic interactions enable the persistent propagation of synchronous activity already in purely random networks without superimposed structures and explain the mechanism underlying it. This study adds a novel perspective on the dynamics of networks with nonlinear interactions in general and presents a new viable mechanism for the occurrence of patterns of precisely timed spikes in recurrent networks.

Ring-shaped neuronal networks: a platform to study persistent activity

Persistent activity in the brain is involved in working memory and motor planning. The ability of the brain to hold information 'online' long after an initiating stimulus is a hallmark of brain areas such as the prefrontal cortex. Recurrent network loops such as the thalamocortical loop and reciprocal loops in the cortex are potential substrates that can support such activity. However, native brain circuitry makes it difficult to study mechanisms underlying such persistent activity. Here we propose a platform to study synaptic mechanisms of such persistent activity by constraining neuronal networks to a recurrent loop like geometry. Using a polymer stamping technique, adhesive proteins are transferred onto glass substrates in a precise ring shape. Primary rat hippocampal cultures were capable of forming ring-shaped networks containing 40-60 neurons. Calcium imaging of these networks show evoked persistent activity in an all-or-none manner. Blocking inhibition with bicuculline methaiodide (BMI) leads to an increase in the duration of persistent activity. These persistent phases were abolished by blockade of asynchronous neurotransmitter release by ethylene glycol tetraacetic acid (EGTA-AM).

BDNF and NT-3 Increase Velocity of Activity Front Propagation in Unidimensional Hippocampal Cultures

Neurotrophins are known to promote synapse development as well as to regulate the efficacy of mature synapses. We have previously reported that in two-dimensional rat hippocampal cultures, brain-derived neurotrophic factor (BDNF) and neurotrophin-3 significantly increase the number of excitatory input connections. Here we measure the effect of these neurotrophic agents on propagating fronts that arise spontaneously in quasi-one-dimensional rat hippocampal cultures. We observe that chronic treatment with BDNF increased the velocity of the propagation front by about 30%. This change is attributed to an increase in the excitatory input connectivity. We analyze the experiment using the Feinerman–Golomb/Ermentrout–Jacobi/Moses–Osan model for the propagation of fronts in a one-dimensional neuronal network with synaptic delay and introduce the synaptic connection probability between adjacent neurons as a new parameter of the model. We conclude that BDNF increases the number of excitatory connections by favoring the probability to form connections between neurons, but without significantly modifying the range of the connections (connectivity footprint).

Beyond the death of linear response: 1/f optimal information transport

Nonergodic renewal processes have recently been shown by several authors to be insensitive to periodic perturbations, thereby apparently sanctioning the death of linear response, a building block of nonequilibrium statistical physics. We show that it is possible to go beyond the "death of linear response" and establish a permanent correlation between an external stimulus and the response of a complex network generating nonergodic renewal processes, by taking as stimulus a similar nonergodic process. The ideal condition of 1/f noise corresponds to a singularity that is expected to be relevant in several experimental conditions.

Complexity and synchronization

We study a fully connected network (cluster) of interacting two-state units as a model of cooperative decision making. Each unit in isolation generates a Poisson process with rate g . We show that when the number of nodes is finite, the decision-making process becomes intermittent. The decision-time distribution density is characterized by inverse power-law behavior with index mu=1.5 and is exponentially truncated. We find that the condition of perfect consensus is recovered by means of a fat tail that becomes more and more extended with increasing number of nodes N . The intermittent dynamics of the global variable are described by the motion of a particle in a double well potential. The particle spends a portion of the total time tau(S) at the top of the potential barrier. Using theoretical and numerical arguments it is proved that tau(S) is proportional to (1/g)ln(const x N) . The second portion of its time, tau(K), is spent by the particle at the bottom of the potential well and it is given by tau(K)=(1/g)exp(const x N) . We show that the time tau(K) is responsible for the Kramers fat tail. This generates a stronger ergodicity breakdown than that generated by the inverse power law without truncation. We establish that the condition of partial consensus can be transmitted from one cluster to another provided that both networks are in a cooperative condition. No significant information transmission is possible if one of the two networks is not yet self-organized. We find that partitioning a large network into a set of smaller interacting clusters has the effect of converting the fat Kramers tail into an inverse power law with mu=1.5 .

Network analysis of biochemical logic for noise reduction and stability: a system of three coupled enzymatic and gates

We develop an approach aimed at optimizing the parameters of a network of biochemical logic gates for reduction of the "analog" noise buildup. Experiments for three coupled enzymatic AND gates are reported, illustrating our procedure. Specifically, starch, one of the controlled network inputs, is converted to maltose by beta-amylase. With the use of phosphate (another controlled input), maltose phosphorylase then produces glucose. Finally, nicotinamide adenine dinucleotide (NAD(+)), the third controlled input, is reduced under the action of glucose dehydrogenase to yield the optically detected signal. Network functioning is analyzed by varying selective inputs and fitting standardized few-parameters "response-surface" functions assumed for each gate. This allows a certain probe of the individual gate quality, but primarily yields information on the relative contribution of the gates to noise amplification. The derived information is then used to modify our experimental system to put it in a regime of a less noisy operation.

Development of input connections in neural cultures

We introduce an approach for the quantitative assessment of the connectivity in neuronal cultures, based on the statistical mechanics of percolation on a graph. This allows us to monitor the development of the culture and to see the emergence of connectivity in the network. The culture becomes fully connected at a time equivalent to the expected time of birth. The spontaneous bursting activity that characterizes cultures develops in parallel with the connectivity. The average number of inputs per neuron can be quantitatively determined in units of m(0), the number of activated inputs needed to excite the neuron. For m(0) approximately 15 we find that hippocampal neurons have on average approximately 60-120 inputs, whereas cortical neurons have approximately 75-150, depending on neuronal density. The ratio of excitatory to inhibitory neurons is determined by using the GABA(A) antagonist bicuculine. This ratio changes during development and reaches the final value at day 7-8, coinciding with the expected time of the GABA switch. For hippocampal cultures the inhibitory cells comprise approximately 30% of the neurons in the culture whereas for cortical cultures they are approximately 20%. Such detailed global information on the connectivity of networks in neuronal cultures is at present inaccessible by any electrophysiological or other technique.

Fast and accurate detection of action potentials from somatic calcium fluctuations

Large-scale recording from a population of neurons is a promising strategy for approaching the study of complex brain functions. Taking advantage of the fact that action potentials reliably evoke transient calcium fluctuations in the cell body, functional multineuron calcium imaging (fMCI) monitors the suprathreshold activity of hundreds of neurons. However, a limitation of fMCI is its semi-manual procedure of spike extraction from somatic calcium fluctuations, which is not only time consuming but is also associated with human errors. Here we describe a novel automatic method that combines principal-component analysis and support vector machine. This simple algorithm determines the timings of the spikes in calcium fluorescence traces more rapidly and reliably than human operators.

Variability and corresponding amplitude-velocity relation of activity propagating in one-dimensional neural cultures

We investigate the propagation of neural activity along one-dimensional rat hippocampal cultures patterned in lines over multielectrode arrays. Activity occurs spontaneously or is evoked by local electrical or chemical stimuli, with different resulting propagation velocities and firing rate amplitudes. A variability of an order of magnitude in velocity and amplitude is observed in spontaneous activity. A linear relation between velocity and amplitude is identified. We define a measure for neuron activation synchrony and find that it correlates with front velocity and is higher for electrically evoked fronts. We present a model that explains the linear relation between amplitude and velocity, which highlights the role of synchrony. The relation to current models for signal propagation in neural media is discussed.

Identification and dynamics of spontaneous burst initiation zones in unidimensional neuronal cultures

Spontaneous activity is typical of in vitro neural networks, often in the form of large population bursts. The origins of this activity are attributed to intrinsically bursting neurons and to noisy backgrounds as well as to recurrent network connections. Spontaneous activity is often observed to emanate from localized sources or initiation zones, propagating from there to excite large populations of neurons. In this study, we use unidimensional cultures to overcome experimental difficulties in identifying initiation zones in vivo and in dissociated two-dimensional cultures. We found that spontaneous activity in these cultures is initiated exclusively in localized zones that are characterized by high neuronal density but also by recurrent and inhibitory network connections. We demonstrate that initiation zones compete in driving network activity in a winner-takes-most scenario.

Percolation in living neural networks

We study living neural networks by measuring the neurons' response to a global electrical stimulation. Neural connectivity is lowered by reducing the synaptic strength, chemically blocking neurotransmitter receptors. We use a graph-theoretic approach to show that the connectivity undergoes a percolation transition. This occurs as the giant component disintegrates, characterized by a power law with an exponent beta approximately or = 0.65. Beta is independent of the balance between excitatory and inhibitory neurons and indicates that the degree distribution is Gaussian rather than scale free.