Predictive haemodynamics in a one-dimensional human carotid artery bifurcation. Part II: application to graft design

A Bayesian surrogate modelling technique is proposed that may be able to predict an optimal bypass graft configuration for patients suffering with stenosis in the internal carotid artery (ICA). At the outset, this statistical technique is considered as a means for identifying key geometric parameters influencing haemodynamics in the human carotid bifurcation. This methodology uses a design of experiments (DoE) technique to generate candidate geometries for flow analysis. A pulsatile one dimensional Navier-Stokes solver incorporating fluid-wall interactions for a Newtonian fluid which predicts pressure and flow in the carotid bifurcation (comprising a stenosed segment in the internal carotid artery) is used for the numerical simulations. Two metrics, pressure variation factor (PVF) and maximum pressure (pm) are employed to directly compare the global and local effects, respectively, of variations in the geometry. The values of PVF and pm are then used to construct two Bayesian surrogate models. These models are statistically analysed to visualise how each geometric parameter influences PVF and pm. Percentage of stenosis is found to influence these pressure based metrics more than any other geometric parameter. Later, we identify bypass grafts with optimal geometric and material properties which have low values of PVF on five test cases with 70%, 75%, 80%, 85% and 90% stenosis in the ICA, respectively

Turbulence modeling in three-dimensional stenosed arterial bifurcations

Under normal healthy conditions, blood flow in the carotid artery bifurcation is laminar. However, in the presence of a stenosis, the flow can become turbulent at the higher Reynolds numbers during systole. There is growing consensus that the transitional k ? model is the best suited Reynolds averaged turbulence model for such flows. Further confirmation of this opinion is presented here by a comparison with the RNG k? model for the flow through a straight, nonbifurcating tube. Unlike similar validation studies elsewhere, no assumptions are made about the inlet profile since the full length of the experimental tube is simulated. Additionally, variations in the inflow turbulence quantities are shown to have no noticeable affect on downstream turbulence intensity, turbulent viscosity, or velocity in the k? model, whereas the velocity profiles in the transitional k? model show some differences due to large variations in the downstream turbulence quantities. Following this validation study, the transitional k? model is applied in a three-dimensional parametrically defined computer model of the carotid artery bifurcation in which the sinus bulb is manipulated to produce mild, moderate, and severe stenosis. The parametric geometry definition facilitates a powerful means for investigating the effect of local shape variation while keeping the global shape fixed. While turbulence levels are generally low in all cases considered, the mild stenosis model produces higher levels of turbulent viscosity and this is linked to relatively high values of turbulent kinetic energy and low values of the specific dissipation rate. The severe stenosis model displays stronger recirculation in the flow field with higher values of vorticity, helicity, and negative wall shear stress. The mild and moderate stenosis configurations produce similar lower levels of vorticity and helicity. DOI: 10.1115/1.240118

Hyperbolic planforms in relation to visual edges and textures perception

We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g. optical imaging, and opens the door to the design of experiments to test these hypotheses. We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific second-order tensor, the structure tensor, perhaps within a hypercolumn. We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. These patterns can be described by what we call hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of the planforms that were used in [1, 2] to account for some visual hallucinations. If these patterns could be observed through brain imaging techniques they would reveal the built-in or acquired invariance of the neural organization to the action of the corresponding subgroups.Comment: 34 pages, 11 figures, 2 table

Democratization in a passive dendritic tree : an analytical investigation

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as “dendritic democracy”. Although well studied experimentally, to date this phenomenon has not been thoroughly explored from a mathematical perspective. In this paper we adopt a passive model of a dendritic tree with distributed excitatory synaptic conductances and analyze a number of key measures of democracy. In particular, via moment methods we derive laws for the transport, from synapse to soma, of strength, characteristic time, and dispersion. These laws lead immediately to synaptic scalings that overcome attenuation with distance. We follow this with a Neumann approximation of Green’s representation that readily produces the synaptic scaling that democratizes the peak somatic voltage response. Results are obtained for both idealized geometries and for the more realistic geometry of a rat CA1 pyramidal cell. For each measure of democratization we produce and contrast the synaptic scaling associated with treating the synapse as either a conductance change or a current injection. We find that our respective scalings agree up to a critical distance from the soma and we reveal how this critical distance decreases with decreasing branch radius

Multiscale Computations on Neural Networks: From the Individual Neuron Interactions to the Macroscopic-Level Analysis

We show how the Equation-Free approach for multi-scale computations can be exploited to systematically study the dynamics of neural interactions on a random regular connected graph under a pairwise representation perspective. Using an individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of simulated annealing we compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level. We also exploit the scheme to perform a rare-events analysis by estimating an effective Fokker-Planck describing the evolving probability density function of the corresponding coarse-grained observables

The role of ongoing dendritic oscillations in single-neuron dynamics

The dendritic tree contributes significantly to the elementary computations a neuron performs while converting its synaptic inputs into action potential output. Traditionally, these computations have been characterized as temporally local, near-instantaneous mappings from the current input of the cell to its current output, brought about by somatic summation of dendritic contributions that are generated in spatially localized functional compartments. However, recent evidence about the presence of oscillations in dendrites suggests a qualitatively different mode of operation: the instantaneous phase of such oscillations can depend on a long history of inputs, and under appropriate conditions, even dendritic oscillators that are remote may interact through synchronization. Here, we develop a mathematical framework to analyze the interactions of local dendritic oscillations, and the way these interactions influence single cell computations. Combining weakly coupled oscillator methods with cable theoretic arguments, we derive phase-locking states for multiple oscillating dendritic compartments. We characterize how the phase-locking properties depend on key parameters of the oscillating dendrite: the electrotonic properties of the (active) dendritic segment, and the intrinsic properties of the dendritic oscillators. As a direct consequence, we show how input to the dendrites can modulate phase-locking behavior and hence global dendritic coherence. In turn, dendritic coherence is able to gate the integration and propagation of synaptic signals to the soma, ultimately leading to an effective control of somatic spike generation. Our results suggest that dendritic oscillations enable the dendritic tree to operate on more global temporal and spatial scales than previously thought

Desynchronization, Mode Locking, and Bursting in Strongly Coupled Integrate-and-Fire Oscillators

This article was published in the journal, Physical Review Letters [© American Physical Society]. It is also available at: http://link.aps.org/abstract/PRL/v81/p2168.We show how a synchronized pair of integrate-and-fire neural oscillators with noninstantaneous synaptic interactions can destabilize in the strong coupling regime resulting in non-phase-locked behavior. In the case of symmetric inhibitory coupling, desynchronization produces an inhomogeneous state in which one of the oscillators becomes inactive (oscillator death). On the other hand, for asymmetric excitatory/inhibitory coupling, mode locking can occur leading to periodic bursting patterns. The consequences for large globally coupled networks is discussed

Linear stability analysis of retrieval state in associative memory neural networks of spiking neurons

We study associative memory neural networks of the Hodgkin-Huxley type of spiking neurons in which multiple periodic spatio-temporal patterns of spike timing are memorized as limit-cycle-type attractors. In encoding the spatio-temporal patterns, we assume the spike-timing-dependent synaptic plasticity with the asymmetric time window. Analysis for periodic solution of retrieval state reveals that if the area of the negative part of the time window is equivalent to the positive part, then crosstalk among encoded patterns vanishes. Phase transition due to the loss of the stability of periodic solution is observed when we assume fast alpha-function for direct interaction among neurons. In order to evaluate the critical point of this phase transition, we employ Floquet theory in which the stability problem of the infinite number of spiking neurons interacting with alpha-function is reduced into the eigenvalue problem with the finite size of matrix. Numerical integration of the single-body dynamics yields the explicit value of the matrix, which enables us to determine the critical point of the phase transition with a high degree of precision.Comment: Accepted for publication in Phys. Rev.

Coordinated optimization of visual cortical maps (I) Symmetry-based analysis

In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), and spatial frequency. It is a long-standing question in theoretical neuroscience whether the observed maps should be interpreted as optima of a specific energy functional that summarizes the design principles of cortical functional architecture. A rigorous evaluation of this optimization hypothesis is particularly demanded by recent evidence that the functional architecture of OP columns precisely follows species invariant quantitative laws. Because it would be desirable to infer the form of such an optimization principle from the biological data, the optimization approach to explain cortical functional architecture raises the following questions: i) What are the genuine ground states of candidate energy functionals and how can they be calculated with precision and rigor? ii) How do differences in candidate optimization principles impact on the predicted map structure and conversely what can be learned about an hypothetical underlying optimization principle from observations on map structure? iii) Is there a way to analyze the coordinated organization of cortical maps predicted by optimization principles in general? To answer these questions we developed a general dynamical systems approach to the combined optimization of visual cortical maps of OP and another scalar feature such as OD or spatial frequency preference.Comment: 90 pages, 16 figure