Michelle Rudolph-Lilith

High-Resolution Intracellular Recordings Using a Real-Time Computational Model of the Electrode

Intracellular recordings of neuronal membrane potential are a central tool in neurophysiology. In many situations, especially in vivo, the traditional limitation of such recordings is the high electrode resistance and capacitance, which may cause significant measurement errors during current injection. We introduce a computer-aided technique, Active Electrode Compensation (AEC), based on a digital model of the electrode interfaced in real time with the electrophysiological setup. The characteristics of this model are first estimated using white noise current injection. The electrode and membrane contribution are digitally separated, and the recording is then made by online subtraction of the electrode contribution. Tests performed in vitro and in vivo demonstrate that AEC enables high-frequency recordings in demanding conditions, such as injection of conductance noise in dynamic-clamp mode, not feasible with a single high-resistance electrode until now. AEC should be particularly useful to characterize fast neuronal phenomena intracellularly in vivo.

Brain networks: small-worlds, after all?

Since its introduction, the ‘small-world’ effect has played a central role in network science, particularly in the analysis of the complex networks of the nervous system. From the cellular level to that of interconnected cortical regions, many analyses have revealed small-world properties in the networks of the brain. In this work, we revisit the quantification of small-worldness in neural graphs. We find that neural graphs fall into the ‘borderline’ regime of small-worldness, residing close to that of a random graph, especially when the degree sequence of the network is taken into account. We then apply recently introducted analytical expressions for clustering and distance measures, to study this borderline small-worldness regime. We derive theoretical bounds for the minimal and maximal small-worldness index for a given graph, and by semi-analytical means, study the small-worldness index itself. With this approach, we find that graphs with small-worldness equivalent to that observed in experimental data are dominated by their random component. These results provide the first thorough analysis suggesting that neural graphs may reside far away from the maximally small-world regime.

Analytical integrate-and-fire neuron models with conductance-based dynamics and realistic postsynaptic potential time course for event-driven simulation strategies

In a previous paper (Rudolph & Destexhe, 2006), we proposed various models, the gIF neuron models, of analytical integrate-and-fire (IF) neurons with conductance-based (COBA) dynamics for use in event-driven simulations. These models are based on an analytical approximation of the differential equation describing the IF neuron with exponential synaptic conductances and were successfully tested with respect to their response to random and oscillating inputs. Because they are analytical and mathematically simple, the gIF models are best suited for fast event-driven simulation strategies. However, the drawback of such models is they rely on a nonrealistic postsynaptic potential (PSP) time course, consisting of a discontinuous jump followed by a decay governed by the membrane time constant. Here, we address this limitation by conceiving an analytical approximation of the COBA IF neuron model with the full PSP time course. The subthreshold and suprathreshold response of this gIF4 model reproduces remarkably well the postsynaptic responses of the numerically solved passive membrane equation subject to conductance noise, while gaining at least two orders of magnitude in computational performance. Although the analytical structure of the gIF4 model is more complex than that of its predecessors due to the necessity of calculating future spike times, a simple and fast algorithmic implementation for use in large-scale neural network simulations is proposed.

On a representation of the Verhulst logistic map

One of the simplest polynomial recursions exhibiting chaotic behavior is the logistic map x n + 1 = a x n ( 1 - x n ) with x n , a ? Q : x n ? 0 , 1 ] ? n ? N and a ? ( 0 , 4 ] , the discrete-time model of the differential growth introduced by Verhulst almost two centuries ago (Verhulst, 1838)? 12]. Despite the importance of this discrete map for the field of nonlinear science, explicit solutions are known only for the special cases a = 2 and a = 4 . In this article, we propose a representation of the Verhulst logistic map in terms of a finite power series in the maps growth parameter a and initial value x 0 whose coefficients are given by the solution of a system of linear equations. Although the proposed representation cannot be viewed as a closed-form solution of the logistic map, it may help to reveal the sensitivity of the map on its initial value and, thus, could provide insights into the mathematical description of chaotic dynamics.

On a Recursive Construction of Circular Paths and the Search for \(\pi \) on the Integer Lattice \(\mathbb {Z}^2\)

Digital circles not only play an important role in various technological settings, but also provide a lively playground for more fundamental number-theoretical questions. In this paper, we present a new algorithm for the construction of digital circles on the integer lattice

Neural graphs: small-worlds, after all?

\mathbb {Z}^2

A discrete algebraic framework for stochastic systems which yield unique and exact solutions

Z2, which makes sole use of the signum function. By briefly elaborating on the nature of discretization of circular paths, we then find that this algorithm recovers, in a space endowed with