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Understanding reliable signal transmission represents a notable challenge for cortical systems, which display a wide range of weights of feedforward and feedback connections among heterogeneous areas. We re-examine the question of signal transmission across the cortex in a network model based on mesoscopic directed and weighted inter-areal connectivity data of the macaque cortex. Our findings reveal that, in contrast to purely feedforward propagation models, the presence of long-range excitatory feedback projections could compromise stable signal propagation. Using population rate models as well as a spiking network model, we find that effective signal propagation can be accomplished by balanced amplification across cortical areas while ensuring dynamical stability. Moreover, the activation of prefrontal cortex in our model requires the input strength to exceed a threshold, which is consistent with the ignition model of conscious processing. These findings demonstrate our model as an anatomically realistic platform for investigations of global primate cortex dynamics.

Reliable signal transmission represents a fundamental challenge for cortical systems, which display a wide range of weights of feedforward and feedback connections among heterogeneous areas. We re-examine the question of signal transmission across the cortex in network models based on recently available mesoscopic, directed-and weighted-inter-areal connectivity data of the macaque cortex. Our findings reveal that, in contrast to feed-forward propagation models, the presence of long-range excitatory feedback projections could compromise stable signal propagation. Using population rate models as well as a spiking network model, we find that effective signal propagation can be accomplished by balanced amplification across cortical areas while ensuring dynamical stability. Moreover, the activation of prefrontal cortex in our model requires the input strength to exceed a threshold, in support of the ignition model of conscious processing, demonstrating our model as an anatomically-realistic platform for investigations of the global primate cortex dynamics.In computational neuroscience, there is a dearth of knowledge about multi-regional brain circuits. New questions that are not crucial for understanding local circuits arise when we investigate how a large-scale brain system works. In particular, reliable signal propagation is a prerequisite for information processing in a hierarchically organized cortical system. A number of studies have been devoted to signal propagation from area to area in the mammalian cortex [1][2][3][4][5][6][7][8]. It was found that a major challenge is to ensure stable transmission, with the signal undergoing neither successive attenuation nor amplification as it travels across multiple areas in a hierarchy. In spite of insights these studies provided, virtually all previous works did not incorporate data-constrained cortical connectivity, and made unrealistic assumptions -for instance, areas are considered identical, network architecture is strictly feedforward, and connection weights are the same at all stages of the hierarchy. Here we argue that achieving stable signal propagation becomes even more challenging with the inclusion of more realistic network architecture and connectivity. Mechanisms that improve signal propagation in simpler models may no longer work for more biological models.Inter-areal cortical network is highly recurrent, abundant with feedback loops [9]. These rich feedback connections pose the risk of destabilizing the system through reverberation as the signal is transmitted across areas. Local cortical circuits are strongly recurrently connected [10], further contributing to system instability. Therefore, mechanisms that improve signal propagation in feedforward networks may quickly lead to instability in a brain-like network. Building such a model requires quantitative anatomical data. Recently, detailed mesoscopic connectivity data has become available for both macaque monkey [9][10][11] and mouse [12,13]. This data delineates a complex inter-areal cortical ...

We investigate the geometry of the edge of chaos for a nine-dimensional sinusoidal shear flow model and show how the shape of the edge of chaos changes with increasing Reynolds number. Furthermore, we numerically compute the scaling of the minimum perturbation required to drive the laminar attracting state into the turbulent region. We find this minimum perturbation to scale with the Reynolds number as Re(-2).

The character of the time-asymptotic evolution of physical systems can have complex, singular behavior with variation of a system parameter, particularly when chaos is involved. A perturbation of the parameter by a small amount can convert an attractor from chaotic to non-chaotic or viceversa. We call a parameter value where this can happen -uncertain. The probability that a random choice of the parameter is -uncertain commonly scales like a power law in . Surprisingly, two seemingly similar ways of defining this scaling, both of physical interest, yield different numerical values for the scaling exponent. We show why this happens and present a quantitative analysis of this phenomenon.While low-dimensional chaotic attractors are common and fundamental in a vast range of physical phenomenon, it is predominantly the case that chaotic motions in such systems are 'structurally unstable'. This typically has the consequence that an arbitrarily small change of a system parameter can always be found that results in periodic behavior [1]. Physical models displaying chaotic attractors that are structurally unstable arise very often, e.g., in studies of plasma dynamics [2], Josephson junctions [3], chemical reactions [4] and many others. We also note that even the simple example of the one-dimensional quadratic map,displays this phenomenon, and, in this paper we will study this example as a convenient paradigm for such situations in general.One reason for concern with this type of behavior is that physical systems have uncertainties in the values of their parameters, and one might therefore ask how confident one can be about the prediction of chaotic behavior from a model calculation (even when the model and its parameter dependence are precisely known and there is no noise). Despite the fundamental importance of this question, very little study has been done to quantitatively address it [5][6][7]. It is the purpose of this paper to readdress this general issue, and, in particular, to resolve a long-standing puzzle. This puzzle has to do with the scaling characterization of the fractal-like chaotic/periodic interweaving structure of parameter dependence associated with structural instability. In particular, studies on the quadratic map Eq.(1) have addressed scaling in two slightly different ways and obtained significantly different estimates of the scaling exponent [5][6][7]. The reason for this surprising discrepancy has remained unresolved. In one of the two ways of addressing scaling, attention was restricted to what might seem to be the most obvious source of uncertainty, namely, when a large (to be defined subsequently) chaotic attractor suddenly turns into a periodic attractor, as a parameter is varied. However, we find such transitions far too rare [5,7] to account for the observed uncertainty, applicable when all chaotic attractors of any size are considered [6], and this observation is at the heart of the resolution of the above-mentioned puzzle.Consider a dynamical system depending on a parameter C, and an attrac...

Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, F : R × M → M, where M is a locally compact manifold without boundary, typically R N . In particular, we investigate F (μ, ·) for μ ∈ J = [μ1, μ2], when F (μ1, ·) has only finitely many periodic orbits while F (μ2, ·) has exponential growth of the number of periodic orbits as a function of the period. For generic F , under additional hypotheses, we use a fixed point index argument to show that there are infinitely many "regular" periodic orbits at μ2. Furthermore, all but finitely many of these regular orbits at μ2 are tethered to their own period-doubling cascade. Specifically, each orbit ρ at μ2 lies in a connected component C(ρ) of regular orbits in J × M; different regular orbits typically are contained in different components, and each component contains a period-doubling cascade. These components are one-manifolds of orbits, meaning that we can reasonably say that an orbit ρ is "tethered" or "tied" to a unique cascade. When F (μ2) has horseshoe dynamics, we show how to count the number of regular orbits of each period, and hence the number of cascades in J × M.As corollaries of our main results, we give several examples, we prove that the map in each example has infinitely many cascades, and we count the cascades.Mathematics Subject Classification (2010). 37G, 37B.

Neuroscience models come in a wide range of scales and specificity, from mean-field rate models to large-scale networks of spiking neurons. There are potential trade-offs between simplicity and realism, versatility and computational speed. This paper is about large-scale cortical network models, and the question we address is one of scalability: would scaling down cell density impact a network’s ability to reproduce cortical dynamics and function? We investigated this problem using a previously constructed realistic model of the monkey visual cortex that is true to size. Reducing cell density gradually up to 50-fold, we studied changes in model behavior. Size reduction without parameter adjustment was catastrophic. Surprisingly, relatively minor compensation in synaptic weights guided by a theoretical algorithm restored mean firing rates and basic function such as orientation selectivity to models 10-20 times smaller than the real cortex. Not all was normal in the reduced model cortices: intracellular dynamics acquired a character different from that of real neurons, and while the ability to relay feedforward inputs remained intact, reduced models showed signs of deficiency in functions that required dynamical interaction among cortical neurons. These findings are not confined to models of the visual cortex, and modelers should be aware of potential issues that accompany size reduction. Broader implications of this study include the importance of homeostatic maintenance of firing rates, and the functional consequences of feedforward versus recurrent dynamics, ideas that may shed light on other species and on systems suffering cell loss.

Attracting chaotic behaviour in dynamical systems is often sensitive to small changes in parameters. If a perturbation in the parameter by a tiny amount ε can change the asymptotic behaviour of the system from being chaotic to being periodic, we call it parameter value ε-uncertain. Here, using a selfsimilar model of the intricate, intertwined parameter-space structure of the chaotic and periodic attractors, we investigate the scaling of this uncertainty with ε. We show that as ε approaches 0, the great majority of ε-uncertain parameters lie in high order windows, that is, windows within windows within windows …. The expected value of the order of the highest order window containing this parameter approaches infinity as ε goes to zero.