1,845 research outputs found
A modified cable formalism for modeling neuronal membranes at high frequencies
Intracellular recordings of cortical neurons in vivo display intense
subthreshold membrane potential (Vm) activity. The power spectral density (PSD)
of the Vm displays a power-law structure at high frequencies (>50 Hz) with a
slope of about -2.5. This type of frequency scaling cannot be accounted for by
traditional models, as either single-compartment models or models based on
reconstructed cell morphologies display a frequency scaling with a slope close
to -4. This slope is due to the fact that the membrane resistance is
"short-circuited" by the capacitance for high frequencies, a situation which
may not be realistic. Here, we integrate non-ideal capacitors in cable
equations to reflect the fact that the capacitance cannot be charged
instantaneously. We show that the resulting "non-ideal" cable model can be
solved analytically using Fourier transforms. Numerical simulations using a
ball-and-stick model yield membrane potential activity with similar frequency
scaling as in the experiments. We also discuss the consequences of using
non-ideal capacitors on other cellular properties such as the transmission of
high frequencies, which is boosted in non-ideal cables, or voltage attenuation
in dendrites. These results suggest that cable equations based on non-ideal
capacitors should be used to capture the behavior of neuronal membranes at high
frequencies.Comment: To appear in Biophysical Journal; Submitted on May 25, 2007; accepted
on Sept 11th, 200
Macroscopic models of local field potentials and the apparent 1/f noise in brain activity
The power spectrum of local field potentials (LFPs) has been reported to
scale as the inverse of the frequency, but the origin of this "1/f noise" is at
present unclear. Macroscopic measurements in cortical tissue demonstrated that
electric conductivity (as well as permittivity) is frequency dependent, while
other measurements failed to evidence any dependence on frequency. In the
present paper, we propose a model of the genesis of LFPs which accounts for the
above data and contradictions. Starting from first principles (Maxwell
equations), we introduce a macroscopic formalism in which macroscopic
measurements are naturally incorporated, and also examine different physical
causes for the frequency dependence. We suggest that ionic diffusion primes
over electric field effects, and is responsible for the frequency dependence.
This explains the contradictory observations, and also reproduces the 1/f power
spectral structure of LFPs, as well as more complex frequency scaling. Finally,
we suggest a measurement method to reveal the frequency dependence of current
propagation in biological tissue, and which could be used to directly test the
predictions of the present formalism
Power-law statistics and universal scaling in the absence of criticality
Critical states are sometimes identified experimentally through power-law
statistics or universal scaling functions. We show here that such features
naturally emerge from networks in self-sustained irregular regimes away from
criticality. In these regimes, statistical physics theory of large interacting
systems predict a regime where the nodes have independent and identically
distributed dynamics. We thus investigated the statistics of a system in which
units are replaced by independent stochastic surrogates, and found the same
power-law statistics, indicating that these are not sufficient to establish
criticality. We rather suggest that these are universal features of large-scale
networks when considered macroscopically. These results put caution on the
interpretation of scaling laws found in nature.Comment: in press in Phys. Rev.
A mean-field model for conductance-based networks of adaptive exponential integrate-and-fire neurons
Voltage-sensitive dye imaging (VSDi) has revealed fundamental properties of
neocortical processing at mesoscopic scales. Since VSDi signals report the
average membrane potential, it seems natural to use a mean-field formalism to
model such signals. Here, we investigate a mean-field model of networks of
Adaptive Exponential (AdEx) integrate-and-fire neurons, with conductance-based
synaptic interactions. The AdEx model can capture the spiking response of
different cell types, such as regular-spiking (RS) excitatory neurons and
fast-spiking (FS) inhibitory neurons. We use a Master Equation formalism,
together with a semi-analytic approach to the transfer function of AdEx
neurons. We compare the predictions of this mean-field model to simulated
networks of RS-FS cells, first at the level of the spontaneous activity of the
network, which is well predicted by the mean-field model. Second, we
investigate the response of the network to time-varying external input, and
show that the mean-field model accurately predicts the response time course of
the population. One notable exception was that the "tail" of the response at
long times was not well predicted, because the mean-field does not include
adaptation mechanisms. We conclude that the Master Equation formalism can yield
mean-field models that predict well the behavior of nonlinear networks with
conductance-based interactions and various electrophysiolgical properties, and
should be a good candidate to model VSDi signals where both excitatory and
inhibitory neurons contribute.Comment: 21 pages, 7 figure
Kramers-Kronig relations and the properties of conductivity and permittivity in heterogeneous media
The macroscopic electric permittivity of a given medium may depend on
frequency, but this frequency dependence cannot be arbitrary, its real and
imaginary parts are related by the well-known Kramers-Kronig relations. Here,
we show that an analogous paradigm applies to the macroscopic electric
conductivity. If the causality principle is taken into account, there exists
Kramers-Kronig relations for conductivity, which are mathematically equivalent
to the Hilbert transform. These relations impose strong constraints that models
of heterogeneous media should satisfy to have a physically plausible frequency
dependence of the conductivity and permittivity. We illustrate these relations
and constraints by a few examples of known physical media. These extended
relations constitute important constraints to test the consistency of past and
future experimental measurements of the electric properties of heterogeneous
media.Comment: 17 pages, 2 figure
Generalized cable formalism to calculate the magnetic field of single neurons and neuronal populations
Neurons generate magnetic fields which can be recorded with macroscopic
techniques such as magneto-encephalography. The theory that accounts for the
genesis of neuronal magnetic fields involves dendritic cable structures in
homogeneous resistive extracellular media. Here, we generalize this model by
considering dendritic cables in extracellular media with arbitrarily complex
electric properties. This method is based on a multi-scale mean-field theory
where the neuron is considered in interaction with a "mean" extracellular
medium (characterized by a specific impedance). We first show that, as
expected, the generalized cable equation and the standard cable generate
magnetic fields that mostly depend on the axial current in the cable, with a
moderate contribution of extracellular currents. Less expected, we also show
that the nature of the extracellular and intracellular media influence the
axial current, and thus also influence neuronal magnetic fields. We illustrate
these properties by numerical simulations and suggest experiments to test these
findings.Comment: Physical Review E (in press); 24 pages, 16 figure
Can power-law scaling and neuronal avalanches arise from stochastic dynamics?
The presence of self-organized criticality in biology is often evidenced by a
power-law scaling of event size distributions, which can be measured by linear
regression on logarithmic axes. We show here that such a procedure does not
necessarily mean that the system exhibits self-organized criticality. We first
provide an analysis of multisite local field potential (LFP) recordings of
brain activity and show that event size distributions defined as negative LFP
peaks can be close to power-law distributions. However, this result is not
robust to change in detection threshold, or when tested using more rigorous
statistical analyses such as the Kolmogorov-Smirnov test. Similar power-law
scaling is observed for surrogate signals, suggesting that power-law scaling
may be a generic property of thresholded stochastic processes. We next
investigate this problem analytically, and show that, indeed, stochastic
processes can produce spurious power-law scaling without the presence of
underlying self-organized criticality. However, this power-law is only apparent
in logarithmic representations, and does not survive more rigorous analysis
such as the Kolmogorov-Smirnov test. The same analysis was also performed on an
artificial network known to display self-organized criticality. In this case,
both the graphical representations and the rigorous statistical analysis reveal
with no ambiguity that the avalanche size is distributed as a power-law. We
conclude that logarithmic representations can lead to spurious power-law
scaling induced by the stochastic nature of the phenomenon. This apparent
power-law scaling does not constitute a proof of self-organized criticality,
which should be demonstrated by more stringent statistical tests.Comment: 14 pages, 10 figures; PLoS One, in press (2010
Modeling extracellular field potentials and the frequency-filtering properties of extracellular space
Extracellular local field potentials (LFP) are usually modeled as arising
from a set of current sources embedded in a homogeneous extracellular medium.
Although this formalism can successfully model several properties of LFPs, it
does not account for their frequency-dependent attenuation with distance, a
property essential to correctly model extracellular spikes. Here we derive
expressions for the extracellular potential that include this
frequency-dependent attenuation. We first show that, if the extracellular
conductivity is non-homogeneous, there is induction of non-homogeneous charge
densities which may result in a low-pass filter. We next derive a simplified
model consisting of a punctual (or spherical) current source with
spherically-symmetric conductivity/permittivity gradients around the source. We
analyze the effect of different radial profiles of conductivity and
permittivity on the frequency-filtering behavior of this model. We show that
this simple model generally displays low-pass filtering behavior, in which fast
electrical events (such as Na-mediated action potentials) attenuate very
steeply with distance, while slower (K-mediated) events propagate over
larger distances in extracellular space, in qualitative agreement with
experimental observations. This simple model can be used to obtain
frequency-dependent extracellular field potentials without taking into account
explicitly the complex folding of extracellular space.Comment: text (LaTeX), 6 figs. (ps
Computing threshold functions using dendrites
Neurons, modeled as linear threshold unit (LTU), can in theory compute all
thresh- old functions. In practice, however, some of these functions require
synaptic weights of arbitrary large precision. We show here that dendrites can
alleviate this requirement. We introduce here the non-Linear Threshold Unit
(nLTU) that integrates synaptic input sub-linearly within distinct subunits to
take into account local saturation in dendrites. We systematically search
parameter space of the nTLU and TLU to compare them. Firstly, this shows that
the nLTU can compute all threshold functions with smaller precision weights
than the LTU. Secondly, we show that a nLTU can compute significantly more
functions than a LTU when an input can only make a single synapse. This work
paves the way for a new generation of network made of nLTU with binary
synapses.Comment: 5 pages 3 figure
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