Structural properties of the Caenorhabditis elegans neuronal network

Despite recent interest in reconstructing neuronal networks, complete wiring diagrams on the level of individual synapses remain scarce and the insights into function they can provide remain unclear. Even for Caenorhabditis elegans, whose neuronal network is relatively small and stereotypical from animal to animal, published wiring diagrams are neither accurate nor complete and self-consistent. Using materials from White et al. and new electron micrographs we assemble whole, self-consistent gap junction and chemical synapse networks of hermaphrodite C. elegans. We propose a method to visualize the wiring diagram, which reflects network signal flow. We calculate statistical and topological properties of the network, such as degree distributions, synaptic multiplicities, and small-world properties, that help in understanding network signal propagation. We identify neurons that may play central roles in information processing, and network motifs that could serve as functional modules of the network. We explore propagation of neuronal activity in response to sensory or artificial stimulation using linear systems theory and find several activity patterns that could serve as substrates of previously described behaviors. Finally, we analyze the interaction between the gap junction and the chemical synapse networks. Since several statistical properties of the C. elegans network, such as multiplicity and motif distributions are similar to those found in mammalian neocortex, they likely point to general principles of neuronal networks. The wiring diagram reported here can help in understanding the mechanistic basis of behavior by generating predictions about future experiments involving genetic perturbations, laser ablations, or monitoring propagation of neuronal activity in response to stimulation.     

Topology of gap junction networks in C. elegans

Gap junctions are prevalent in every nervous system, but their role in information processing remains largely unknown. In C. elegans, the role of gap junctional communication in touch sensitivity has been demonstrated. In this animal, the entire complement of gap junctions in the nervous system is documented, therefore providing a good model for the computational investigation of circuit functions of gap junctions.We explored several hypotheses about the role of gap junctions in the nervous system of C. elegans by systematically analysing an anatomical database with recursive algorithms. We find that gap junctions connect different sets of neurons from those connected by chemical synapses. In addition, when analysing the topology of the gap-junction networks, we find that, surprisingly, most (92%) neurons in the worm are linked in a single gap-junction network. The worm nervous system can only be divided into smaller networks by assuming that two or more gap junctions are necessary for functional coupling or that neural activity has limited propagation. However, these groups, and others identified using algorithms with subsets or combinations of restrictive criteria, do not correspond to any known circuits identified in genetic and behavioral studies. Finally, we notice that the function of some gap junctions appears linked to their precise location on the neuronal processes. We propose that the location of the gap junctions within the neuron determines their functional role.     

Modelling the Effects of Electrical Coupling between Unmyelinated Axons of Brainstem Neurons Controlling Rhythmic Activity

Gap junctions between fine unmyelinated axons can electrically couple groups of brain neurons to synchronise firing and contribute to rhythmic activity. To explore the distribution and significance of electrical coupling, we modelled a well analysed, small population of brainstem neurons which drive swimming in young frog tadpoles. A passive network of 30 multicompartmental neurons with unmyelinated axons was used to infer that: axon-axon gap junctions close to the soma gave the best match to experimentally measured coupling coefficients; axon diameter had a strong influence on coupling; most neurons were coupled indirectly via the axons of other neurons. When active channels were added, gap junctions could make action potential propagation along the thin axons unreliable. Increased sodium and decreased potassium channel densities in the initial axon segment improved action potential propagation. Modelling suggested that the single spike firing to step current injection observed in whole-cell recordings is not a cellular property but a dynamic consequence of shunting resulting from electrical coupling. Without electrical coupling, firing of the population during depolarising current was unsynchronised; with coupling, the population showed synchronous recruitment and rhythmic firing. When activated instead by increasing levels of modelled sensory pathway input, the population without electrical coupling was recruited incrementally to unpatterned activity. However, when coupled, the population was recruited all-or-none at threshold into a rhythmic swimming pattern: the tadpole “decided” to swim. Modelling emphasises uncertainties about fine unmyelinated axon physiology but, when informed by biological data, makes general predictions about gap junctions: locations close to the soma; relatively small numbers; many indirect connections between neurons; cause of action potential propagation failure in fine axons; misleading alteration of intrinsic firing properties. Modelling also indicates that electrical coupling within a population can synchronize recruitment of neurons and their pacemaker firing during rhythmic activity.     

Gap junctions and neurological disorders of the central nervous system

Gap junctions are intercellular channels which directly connect the cytoplasm between neighboring cells. In the central nervous system (CNS) various kinds of cells are coupled by gap junctions, which play an important role in maintaining normal function. Neuronal gap junctions are involved in electrical coupling and may also contribute to the recovery of function after cell injury. Astrocytes are involved in the pathology of most neuronal disorders, including brain ischemia, Alzheimer's disease and epilepsy. In the pathology of brain tumors, gap junctions may be related to the degree of malignancy and metastasis. However, the role of connexins, gap junctions and hemichannels in the pathology of the diseases in the CNS is still ambiguous. Of increasing importance is the unraveling of the function of gap junctions in the neural cell network, involving neurons, astrocytes, microglia and oligodendrocytes. A better understanding of the role of gap junctions may contribute to the development of new therapeutic approaches to treating diseases of the CNS.     

Electrical Transmission between Mammalian Neurons is Supported by a Small Fraction of Gap Junction Channels

Electrical synapses formed by gap junctions between neurons create networks of electrically coupled neurons in the mammalian brain, where these networks have been found to play important functional roles. In most cases, interneuronal gap junctions occur at remote dendro-dendritic contacts, making difficult accurate characterization of their physiological properties and correlation of these properties with their anatomical and morphological features of the gap junctions. In the mesencephalic trigeminal (MesV) nucleus where neurons are readily accessible for paired electrophysiological recordings in brain stem slices, our recent data indicate that electrical transmission between MesV neurons is mediated by connexin36 (Cx36)-containing gap junctions located at somato-somatic contacts. We here review evidence indicating that electrical transmission between these neurons is supported by a very small fraction of the gap junction channels present at cell-cell contacts. Acquisition of this evidence was enabled by the unprecedented experimental access of electrical synapses between MesV neurons, which allowed estimation of the average number of open channels mediating electrical coupling in relation to the average number of gap junction channels present at these contacts. Our results indicate that only a small proportion of channels (~0.1?%) appear to be conductive. On the basis of similarities with other preparations, we postulate that this phenomenon might constitute a general property of vertebrate electrical synapses, reflecting essential aspects of gap junction function and maintenance.     

Reconstructing the three-dimensional GABAergic microcircuit of the striatum

A system's wiring constrains its dynamics, yet modelling of neural structures often overlooks the specific networks formed by their neurons. We developed an approach for constructing anatomically realistic networks and reconstructed the GABAergic microcircuit formed by the medium spiny neurons (MSNs) and fast-spiking interneurons (FSIs) of the adult rat striatum. We grew dendrite and axon models for these neurons and extracted probabilities for the presence of these neurites as a function of distance from the soma. From these, we found the probabilities of intersection between the neurites of two neurons given their inter-somatic distance, and used these to construct three-dimensional striatal networks. The MSN dendrite models predicted that half of all dendritic spines are within 100μm of the soma. The constructed networks predict distributions of gap junctions between FSI dendrites, synaptic contacts between MSNs, and synaptic inputs from FSIs to MSNs that are consistent with current estimates. The models predict that to achieve this, FSIs should be at most 1% of the striatal population. They also show that the striatum is sparsely connected: FSI-MSN and MSN-MSN contacts respectively form 7% and 1.7% of all possible connections. The models predict two striking network properties: the dominant GABAergic input to a MSN arises from neurons with somas at the edge of its dendritic field; and FSIs are inter-connected on two different spatial scales: locally by gap junctions and distally by synapses. We show that both properties influence striatal dynamics: the most potent inhibition of a MSN arises from a region of striatum at the edge of its dendritic field; and the combination of local gap junction and distal synaptic networks between FSIs sets a robust input-output regime for the MSN population. Our models thus intimately link striatal micro-anatomy to its dynamics, providing a biologically grounded platform for further study.     

A novel form of cellular communication among thymic epithelial cells: intercellular calcium wave propagation

We here describe intercellular calcium waves as a novel form of cellular communication among thymic epithelial cells. We first characterized the mechanical induction of intercellular calcium waves in different thymic epithelial cell preparations: cortical 1-4C18 and medullary 3-10 thymic epithelial cell lines and primary cultures of thymic "nurse" cells. All thymic epithelial preparations responded with intercellular calcium wave propagation after mechanical stimulation. In general, the propagation efficacy of intercellular calcium waves in these cells was high, reaching 80-100% of the cells within a given confocal microscopic field, with a mean velocity of 6-10 µm/s and mean amplitude of 1.4- to 1.7-fold the basal calcium level. As evaluated by heptanol and suramin treatment, our results suggest the participation of both gap junctions and P2 receptors in the propagation of intercellular calcium waves in thymic nurse cells and the more prominent participation of gap junctions in thymic epithelial cell lines. Finally, in cocultures, the transmission of intercellular calcium wave was not observed between the mechanically stimulated thymic epithelial cell and adherent thymocytes, suggesting that intercellular calcium wave propagation is limited to thymic epithelial cells and does not affect the neighboring thymocytes. In conclusion, these data describe for the first time intercellular calcium waves in thymic epithelial cells and the participation of both gap junctions and P2 receptors in their propagation. gap junctions; connexin43; P2 receptors; intercellular communication     

THE APPEARANCE AND STRUCTURE OF INTERCELLULAR CONNECTIONS DURING THE ONTOGENY OF THE RABBIT OVARIAN FOLLICLE WITH PARTICULAR REFERENCE TO GAP JUNCTIONS

Lanthanum tracer and freeze-fracture electron microscope techniques were used to study junctional complexes between granulosa cells during the differentiation of the rabbit ovarian follicle. For convenience we refer to cells encompassing the oocyte, before antrum and gap junction formation, as follicle cells. After the appearance of an antrum and gap junctions we call the cells granulosa cells. Maculae adherentes are found at the interfaces of oocyte-follicle-granulosa cells throughout folliculogenesis. Gap junctions are first detected in follicles when the antrum appears. In early antral follicles typical large gap junctions are randomly distributed between granulosa cells. In freeze-fracture replicas, they are characterized by polygonally packed 90-Å particles arranged in rows separated by nonparticulate A-face membrane. A particle-sparse zone surrounds gap junctions and is frequently occupied by small particle aggregates of closely packed intramembranous particles. The gap junctions of granulosa cells appear to increase in size with further differentiation of the follicle. The granulosa cells of large Graafian follicles are adjoined by small and large gap junctions; annular gap junctions are also present. The large gap junctions are rarely surrounded by a particle-free zone on their A-faces, but are further distinguished by particle rows displaying a higher degree of organization.     

Calcium waves in astrocytes-filling in the gaps.

Stimulus-evoked cellular responses are sometimes organized in the form of propagating waves of cytoplasmic Ca2+ increase. Ca2+ waves can be elicited in cultured astrocytes by the neurotransmitter glutamate; however, the propagation mechanism is unknown. Here, qualitative and quantitative features of propagation suggest that astrocytic Ca2+ waves are mediated by an intracellular signal that crosses intercellular junctions. The role of gap junctions in cell-cell Ca2+ wave propagation was specifically tested. Functional gap junctions were demonstrated using a noninvasive fluorescence recovery method and the gap junction blockers halothane and octanol. Gap junction closure prevented intracellular waves from propagating between cells without affecting the velocity of the intracellular wave itself. The pivotal role played by the gap junction creates the potential for dynamic changes in glial connectivity and long-range glial signaling.     

Detecting and estimating rectification of gap junction conductance based on simulations of dual-cell recordings from a pair and a network of coupled cells

Gap junctions can exhibit rectification of conductance. Some reports use inequality of coupling coefficients as the first sign of the possible existence of rectification ( [Devor and Yarom, 2002], [Fan et al., 2005], [31], [Mann-Metzer and Yarom, 1999], [Nolan et al., 1999] and [Szabadics et al., 2001]). However, mathematical modeling and simulations of electrotonic coupling between an isolated pair of neurons showed conditions where the coupling coefficients were unreliable indicators of rectification. On the other hand, the transfer resistances were found to be reliable indicators of junctional rectification. The existing mathematical model of cell coupling ( [Bennett, 1966], [Devor and Yarom, 2002] and [Verselis and Veenstra, 2000]) was extended in order to measure rectification of the junctional conductances directly between dual-recorded neurons whether isolated or surrounded by a simulated 3-dimensional network of heterogeneous cells whose gap junctions offered parallel paths for current flow between the recorded neurons. The results showed that the transfer resistances could still detect rectification of the gap junction linking the dual-recorded neurons when embedded in a coupled cell network and that a mathematical model could estimate the conductances in both directions through this gap junction using only data that would be available from real dual-intracellular penetrations which allow electrophysiological recordings and intracellular staining. Rectification of gap junctions in unrecorded cells of a biologically realistic coupled cell network had negligible effects on the voltage responses of the dual-recorded neurons because of minimal current passing through these surrounding cells.     

Axo-axonal coupling. a novel mechanism for ultrafast neuronal communication

We provide physiological, pharmacological, and structural evidence that axons of hippocampal principal cells are electrically coupled, with prepotentials or spikelets forming the physiological substrate of electrical coupling as observed in cell somata. Antidromic activation of neighboring axons induced somatic spikelet potentials in neurons of CA3, CA1, and dentate gyrus areas of rat hippocampal slices. Somatic invasion by these spikelets was dependent on the activation of fast Na(+) channels in the postjunctional neuron. Antidromically elicited spikelets were suppressed by gap junction blockers and low intracellular pH. Paired axo-somatic and somato-dendritic recordings revealed that the coupling potentials appeared in the axon before invading the soma and the dendrite. Using confocal laser scanning microscopy we found that putative axons of principal cells were dye coupled. Our data thus suggest that hippocampal neurons are coupled by axo-axonal junctions, providing a novel mechanism for very fast electrical communication.     

Potassium diffusive coupling in neural networks

Conventional neural networks are characterized by many neurons coupled together through synapses. The activity, synchronization, plasticity and excitability of the network are then controlled by its synaptic connectivity. Neurons are surrounded by an extracellular space whereby fluctuations in specific ionic concentration can modulate neuronal excitability. Extracellular concentrations of potassium ([K⁺]_o) can generate neuronal hyperexcitability. Yet, after many years of research, it is still unknown whether an elevation of potassium is the cause or the result of the generation, propagation and synchronization of epileptiform activity. An elevation of potassium in neural tissue can be characterized by dispersion (global elevation of potassium) and lateral diffusion (local spatial gradients). Both experimental and computational studies have shown that lateral diffusion is involved in the generation and the propagation of neural activity in diffusively coupled networks. Therefore, diffusion-based coupling by potassium can play an important role in neural networks and it is reviewed in four sections. Section 2 shows that potassium diffusion is responsible for the synchronization of activity across a mechanical cut in the tissue. A computer model of diffusive coupling shows that potassium diffusion can mediate communication between cells and generate abnormal and/or periodic activity in small (§3) and in large networks of cells (§4). Finally, in §5, a study of the role of extracellular potassium in the propagation of axonal signals shows that elevated potassium concentration can block the propagation of neural activity in axonal pathways. Taken together, these results indicate that potassium accumulation and diffusion can interfere with normal activity and generate abnormal activity in neural networks.     

Nonlinear Gap Junctions Enable Long-Distance Propagation of Pulsating Calcium Waves in Astrocyte Networks

A new paradigm has recently emerged in brain science whereby communications between glial cells and neuron-glia interactions should be considered together with neurons and their networks to understand higher brain functions. In particular, astrocytes, the main type of glial cells in the cortex, have been shown to communicate with neurons and with each other. They are thought to form a gap-junction-coupled syncytium supporting cell-cell communication via propagating Ca²⁺ waves. An identified mode of propagation is based on cytoplasm-to-cytoplasm transport of inositol trisphosphate (IP₃) through gap junctions that locally trigger Ca²⁺ pulses via IP₃-dependent Ca²⁺-induced Ca²⁺ release. It is, however, currently unknown whether this intracellular route is able to support the propagation of long-distance regenerative Ca²⁺ waves or is restricted to short-distance signaling. Furthermore, the influence of the intracellular signaling dynamics on intercellular propagation remains to be understood. In this work, we propose a model of the gap-junctional route for intercellular Ca²⁺ wave propagation in astrocytes. Our model yields two major predictions. First, we show that long-distance regenerative signaling requires nonlinear coupling in the gap junctions. Second, we show that even with nonlinear gap junctions, long-distance regenerative signaling is favored when the internal Ca²⁺ dynamics implements frequency modulation-encoding oscillations with pulsating dynamics, while amplitude modulation-encoding dynamics tends to restrict the propagation range. As a result, spatially heterogeneous molecular properties and/or weak couplings are shown to give rise to rich spatiotemporal dynamics that support complex propagation behaviors. These results shed new light on the mechanisms implicated in the propagation of Ca²⁺ waves across astrocytes and the precise conditions under which glial cells may participate in information processing in the brain.     

Asymmetric temporal properties in the receptive field of retinal transient amacrine cells

The speed of signal conduction is a factor determining the temporal properties of individual neurons and neuronal networks. We observed very different conduction velocities within the receptive field of fast-type On-Off transient amacrine cells in carp retina cells, which are tightly coupled to each other via gap junctions. The fastest speeds were found in the dorsal area of the receptive fields, on average five times faster than those detected within the ventral area. The asymmetry was similar in the On- and Off-part of the responses, thus being independent of the pathway, pointing to the existence of a functional mechanism within the recorded cells themselves. Nonetheless, the spatial decay of the graded-voltage photoresponse within the receptive field was found to be symmetrical, with the amplitude center of the receptive field being displaced to the faster side from the minimum-latency location. A sample of the orientation of varicosity-laden polyaxons in neurobiotin-injected cells supported the model, revealing that approximately 75% of these processes were directed dorsally from the origin cells. Based on these results, we modeled the velocity asymmetry and the displacement of amplitude center by adding a contribution of an asymmetric polyaxonal inhibition to the network. Due to the asymmetry in the conduction velocity, the time delay of a light response is proposed to depend on the origin of the photostimulus movement, a potentially important mechanism underlying direction selectivity within the inner retina.     

High-frequency oscillations and sequence generation in two-population models of hippocampal region CA1

Hippocampal sharp wave/ripple oscillations are a prominent pattern of collective activity, which consists of a strong overall increase of activity with superimposed (140 − 200 Hz) ripple oscillations. Despite its prominence and its experimentally demonstrated importance for memory consolidation, the mechanisms underlying its generation are to date not understood. Several models assume that recurrent networks of inhibitory cells alone can explain the generation and main characteristics of the ripple oscillations. Recent experiments, however, indicate that in addition to inhibitory basket cells, the pattern requires in vivo the activity of the local population of excitatory pyramidal cells. Here, we study a model for networks in the hippocampal region CA1 incorporating such a local excitatory population of pyramidal neurons. We start by investigating its ability to generate ripple oscillations using extensive simulations. Using biologically plausible parameters, we find that short pulses of external excitation triggering excitatory cell spiking are required for sharp/wave ripple generation with oscillation patterns similar to in vivo observations. Our model has plausible values for single neuron, synapse and connectivity parameters, random connectivity and no strong feedforward drive to the inhibitory population. Specifically, whereas temporally broad excitation can lead to high-frequency oscillations in the ripple range, sparse pyramidal cell activity is only obtained with pulse-like external CA3 excitation. Further simulations indicate that such short pulses could originate from dendritic spikes in the apical or basal dendrites of CA1 pyramidal cells, which are triggered by coincident spike arrivals from hippocampal region CA3. Finally we show that replay of sequences by pyramidal neurons and ripple oscillations can arise intrinsically in CA1 due to structured connectivity that gives rise to alternating excitatory pulse and inhibitory gap coding; the latter denotes phases of silence in specific basket cell groups, which induce selective disinhibition of groups of pyramidal neurons. This general mechanism for sequence generation leads to sparse pyramidal cell and dense basket cell spiking, does not rely on synfire chain-like feedforward excitation and may be relevant for other brain regions as well.     

Interpreting voltage-sensitivity of gap junctions as a mechanism of cardiac memory

Memory in the nervous system is essentially a network effect, resulting from activity-dependent synaptic modification in a network of neurons. Like the nervous system, the heart is a network of cardiac cells electrically coupled by gap junctions. The heart too has memory, termed cardiac memory, whereby the effect of an external electrical activation persists long after the presentation of stimulus is terminated. We have earlier proposed that adaptation of gap junctions, as a function of membrane voltages of the cells that are coupled by the gap junctions, is related to cardiac memory [V.S. Chakravarthy, J. Ghosh, On Hebbian-like adaption in heart muscle: a proposal for "Cardiac Memory", Biol. Cybern. 76 (1997) 207, J. Krishnan, V.S. Chakravarthy, S. Radhakrishnan, On the role of gap junctions on cardiac memory effect, Comput. Cardiol. 32 (2005) 13]. Using the proposed mechanism, we demonstrate memory effect using computational models of interacting cell pairs. In this paper, we address the biological validity of the proposed mechanism of gap junctional adaptation. It is known from electrophysiology of gap junctions that the conductance of these channels adapts as a function of junctional voltage. At a first sight, this form of voltage dependence seems to be at variance with the form required by our mechanism. But we show, with the help of a theoretical model, that the proposed mechanism of voltage-dependent adaptation of gap junctions, is compatible with the known voltage-sensitivity of gap junctions observed in electrophysiological studies. Our analysis suggests a new significance of the voltage-sensitivity of gap junctions and its possible link to the phenomenon of cardiac memory.     

Gap Junctions Contribute to the Regulation of Walking-Like Activity in the Adult Mudpuppy (Necturus Maculatus)

Although gap junctions are widely expressed in the developing central nervous system, the role of electrical coupling of neurons and glial cells via gap junctions in the spinal cord in adults is largely unknown. We investigated whether gap junctions are expressed in the mature spinal cord of the mudpuppy and tested the effects of applying gap junction blocker on the walking-like activity induced by NMDA or glutamate in an in vitro mudpuppy preparation. We found that glial and neural cells in the mudpuppy spinal cord expressed different types of connexins that include connexin 32 (Cx32), connexin 36 (Cx36), connexin 37 (Cx37), and connexin 43 (Cx43). Application of a battery of gap junction blockers from three different structural classes (carbenexolone, flufenamic acid, and long chain alcohols) substantially and consistently altered the locomotor-like activity in a dose-dependent manner. In contrast, these blockers did not significantly change the amplitude of the dorsal root reflex, indicating that gap junction blockers did not inhibit neuronal excitability nonselectively in the spinal cord. Taken together, these results suggest that gap junctions play a significant modulatory role in the spinal neural networks responsible for the generation of walking-like activity in the adult mudpuppy.     

Mechanisms of Seizure Propagation in 2-Dimensional Centre-Surround Recurrent Networks

Understanding how seizures spread throughout the brain is an important problem in the treatment of epilepsy, especially for implantable devices that aim to avert focal seizures before they spread to, and overwhelm, the rest of the brain. This paper presents an analysis of the speed of propagation in a computational model of seizure-like activity in a 2-dimensional recurrent network of integrate-and-fire neurons containing both excitatory and inhibitory populations and having a difference of Gaussians connectivity structure, an approximation to that observed in cerebral cortex. In the same computational model network, alternative mechanisms are explored in order to simulate the range of seizure-like activity propagation speeds (0.1–100 mm/s) observed in two animal-slice-based models of epilepsy: (1) low extracellular , which creates excess excitation and (2) introduction of gamma-aminobutyric acid (GABA) antagonists, which reduce inhibition. Moreover, two alternative connection topologies are considered: excitation broader than inhibition, and inhibition broader than excitation. It was found that the empirically observed range of propagation velocities can be obtained for both connection topologies. For the case of the GABA antagonist model simulation, consistent with other studies, it was found that there is an effective threshold in the degree of inhibition below which waves begin to propagate. For the case of the low extracellular model simulation, it was found that activity-dependent reductions in inhibition provide a potential explanation for the emergence of slowly propagating waves. This was simulated as a depression of inhibitory synapses, but it may also be achieved by other mechanisms. This work provides a localised network understanding of the propagation of seizures in 2-dimensional centre-surround networks that can be tested empirically.     

Robustness of Oscillatory Behavior in Correlated Networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.