Figure 1.
Schematic illustration of the input-output relationship between transmembrane currents (input) and the different measurement modalities (output).
The transmembrane currents are illustrated by synaptic currents and channel currents. A synaptic current is commonly modeled by means of exponentially decaying functions (synaptic kernel) triggered by incoming spike trains, whereas a channel current typically is modeled by a channel switching between an open state (o), letting a current with constant amplitude through the channel, or a closed state (c). The input currents are filtered by the neuronal cable, resulting in a low-pass filtered output current in the soma with a power spectral density (PSD) designated . The PSDs of the other measurement modalities studied here, i.e., the soma potential (
) and the current-dipole moment giving the single-neuron contribution to the EEG (
), are typically even more low-pass filtered, as illustrated by the PSDs plotted in the lower right panel.
Figure 2.
Normalized power spectral densities (PSDs) for the soma current, the current-dipole moment (i.e., EEG contribution) and the soma potential for a ball and stick neuron and a pyramidal neuron.
A homogeneous density of noisy input currents is applied throughout the neural membrane. Columns 1 (ball and stick neuron) and 2 (pyramidal neuron) show PSDs for white-noise input, the blue and green lines correspond to uncorrelated and correlated input currents, respectively. Note that there is no green line in the two upper rows, since a homogeneous density of correlated inputs throughout the neuron gives no net soma current or dipole moment. An ensemble of PSDs from 20 single input currents for the ball and stick neuron and 107 single input currents for the pyramidal neuron is shown in grey. The results for the most distal synapses are shown in dark grey and the results for the proximal synapses in light grey, corresponding to the color shown in the filled circles at the respective neuron morphology (between columns 1 and 2). Column 3 illustrates how different power-law spectra of the input currents change the output PSDs: the blue, pink and brown lines express the PSD for uncorrelated white (constant), pink () and Brownian noise input (
), respectively. The values of
in legends denote estimated power-law exponents at 1000 Hz, i.e., the negative discrete log-log derivative,
. In the rightmost column the values of
correspond to pink noise input, for Brownian noise input and white-noise input the values are ‘+1’ and ‘−1’ with respect to the pink input, respectively, as indicated by the brown ‘+’ and the blue ‘−’. The ball and stick neuron was simulated with 200 dendritic segments (corresponding to the default parameters listed in Table 1), while the pyramidal neuron was simulated with 3214 dendritic segments. Broken lines correspond to the ball and stick neuron, whole lines to the pyramidal neuron.
Table 1.
List of symbols in alphabetical order.
Figure 3.
Schematic illustration of the ball and stick neuron model and its filtering properties.
(A) Schematic illustration of the ball and stick neuron model with a single input at a given position . The lumped soma is assumed iso-potential and located at
. (B) Frequency-dependent current-density envelopes of return currents for a ball and stick neuron with input at
. The somatic return currents are illustrated as current densities from a soma section with length
placed below the stick. For 1 Hz, 10 Hz, 100 Hz and 1000 Hz the amplitudes of the somatic return currents are about 1/7.3, 1/7.5, 1/22 and 1/3100 of the input current, respectively. Parameters used for the ball and stick neuron model: stick diameter
, somatic diameter
, stick length
mm, specific membrane resistance
, inner resistivity
m and specific membrane capacitance of
. This parameter set is the default parameter set used in the present study, see Table 1. (C) Representative log-log plot for a PSD when input is homogeneously distributed across the entire neuron model. The low frequency (lf), intermediate frequency (if) and high frequency (hf) regimes are stipulated. The regimes are defined relatively to
describing the asymptotic value of the exponent of the respective power-law transfer functions (
,
or
), with both uncorrelated and correlated input (‘all’ types of input) onto both the soma and the stick.
Table 2.
PSD amplitudes and high-frequency power laws.
Figure 4.
Slopes for the PSD transfer function for the soma current for a ball and stick neuron in terms of its dimensionless parameters.
Row 1 corresponds both to correlated input currents () with any input densities
, and to uncorrelated input to soma only (
). Row 2 corresponds to the case of uncorrelated input currents solely onto the dendrite. Row 3 corresponds to uncorrelated input currents with equal density,
, throughout the neuron. The dimensionless parameter
is plotted along the vertical axes, while the dimensionless frequency
is plotted logarithmically along the horizontal axes. In the left column the dimensionless length is
, in the middle column
and the right column
. The horizontal white line express the default value of the parameter
,
(soma diameter
, stick diameter
, length constant
mm), while the vertical white lines correspond to frequencies of 10 Hz, 100 Hz and 1000 Hz, respectively, for the default membrane time constant
ms. The thin black line denotes
and the thicker black line denotes
, with
denoting the asymptotic value for the case of both uncorrelated and correlated input onto the whole neuron. All plots use the same color scale for
, given by the color bar to the right.
Figure 5.
Slopes for the PSD transfer function for the current-dipole moment (single-neuron EEG contribution) for a ball and stick neuron in terms of its dimensionless parameters.
Row 1 corresponds both to correlated input currents () with any input densities
, and to uncorrelated input to soma only (
). Row 2 corresponds to the case of input currents solely onto the dendrite. Row 3 corresponds to uncorrelated white-noise input currents with equal density,
, throughout the neuron. The dimensionless parameter
is plotted along the vertical axes, while the dimensionless frequency
is plotted logarithmically along the horizontal axes. In the left column the dimensionless length is
, in the middle column
and the right column
. The horizontal white line express the default value of the parameter
,
(soma diameter
, stick diameter
, length constant
mm), while the vertical white lines correspond to frequencies of 10 Hz, 100 Hz and 1000 Hz for the default membrane time constant
ms. The thin black line denotes
and the thicker black line denotes
, with
denoting the asymptotic value for the case of both uncorrelated and correlated input onto the whole neuron. All plots use the same color scale for
, given by the color bar to the right.
Figure 6.
Slopes for the PSD transfer function for the soma potential for a ball and stick neuron in terms of its dimensionless parameters.
Row 1 corresponds to correlated input currents solely onto the dendrite. Row 2 corresponds to input currents solely onto soma, either correlated () or uncorrelated (
). In row 3 uncorrelated input currents are applied homogeneously across the dendrite. Row 5 corresponds to uncorrelated input currents with equal density,
, throughout the neuron. Row 6 shows results for correlated input currents with equal density,
, throughout the neuron. The dimensionless parameter
is plotted along the vertical axes, while the dimensionless frequency
is plotted logarithmically along the horizontal axes. In the left column the dimensionless length is
, in the middle column
and the right column
. The horizontal white line express the default value of the parameter
,
(soma diameter
, stick diameter
, length constant
mm), while the vertical white lines correspond to frequencies of 10 Hz, 100 Hz and 1000 Hz for the default membrane time constant
ms. The thin black line denotes
and the thicker black line denotes
, with
denoting the asymptotic value for the case of both uncorrelated and correlated input onto the whole neuron. All plots use the same color scale for
, given by the color bar to the right.
Figure 7.
Dependence of PSDs on biophysical parameters for uncorrelated input.
PSDs of the soma current (row 1), current-dipole moment (row 2) and soma potential (row 3) for the ball and stick model with uncorrelated white-noise input currents homogeneously distributed throughout the membrane. The input density is two inputs per square micrometer, and the input current is assumed to have a constant (white noise) PSD, . The columns show variation with stick length (first column), specific membrane resistance (second column), stick diameter (third column) and soma diameter (fourth column) with values shown in the legends below the panels. All other parameters of the ball and stick neuron have default values: stick diameter
, somatic diameter
, stick length
mm, specific membrane resistance
, inner resistivity
m and a specific membrane capacitance
. The values of
printed in the legends describe the powers of the slopes at 1000 Hz. The upper
corresponds to the low value of the parameter varied (green), the middle
corresponds to the default parameter (red), while the lower
corresponds to the high value of the parameter varied (blue).
Figure 8.
Dependence of PSDs on biophysical parameters for correlated input.
PSDs of the soma current (row 1), current-dipole moment (row 2) and soma potential (rows 3 to 5) for the ball and stick model with correlated white-noise input currents homogeneously distributed throughout the stick only (row 1 to 3), the soma only (row 5) or with equal density throughout the soma and the stick (row 4). The input density is two inputs per square micrometer, unless a zero density is indicated on the axis. The input current is assumed to have a constant (white noise) PSD, . The columns show variation with stick length (first column), specific membrane resistance (second column), stick diameter (third column) and soma diameter (fourth column) with values shown in the legends below the panels. All other parameters of the ball and stick neuron have default values: stick diameter
, somatic diameter
, stick length
mm, specific membrane resistance
, inner resistivity
m and a specific membrane capacitance
. The values of
printed in the legends describe the powers of the slopes at 1000 Hz. The upper
corresponds to the low value of the parameter varied (green), the middle
corresponds to the default parameter (red), while the lower
corresponds to the high value of the parameter varied (blue).
Figure 9.
Suggested scenario for generation of soma-potential noise in the in vivo situation with a combination of membrane current sources, presumably due to intrinsic ion channels, and synaptic current sources.
Both sources are assumed uncorrelated and homogeneously spread out across a ball and stick neuron. (A) Excerpt of real-time soma potential following injection of synaptic noise through an exponential synapse (white noise filtered through Eq. (117), blue line), noise, putatively from intrinsic ion channel (white noise filtered through a
filter, red line), and sum of both (black line). (B) Histogram over soma potential for the three situations in A (50 s period with a sampling rate of 10 kHz). (C) Soma-potential PSDs for five cases: the three cases in A (
; exponential synapse, Eq. (117); sum of
and exponential synapse) as well as alpha-function synapse (Eq. 118, green line) and sum of alpha-function synapse and
(magenta line). All traces are normalized to the value of the summed PSDs for
noise and exponential synapse for the lowest depicted frequency (0.1 Hz). (D) Locally (in frequency) estimated power-law coefficient
, i.e., Eq. (116). The noise amplitudes are set so that soma-potential noise from (i) the
current noise input has a standard deviation of
= 0.6 mV (as seen in in vitro experiments [19]; frequencies between 0.2 and 100 Hz included in the noise variance sum) and (ii) total noise (synaptic+
) a standard deviation of
= 2.5 mV (similar to in vivo experiments reported in Fig. 11 in [18]). Parameters used for the ball and stick neuron model is the default values (cf. caption of Fig. 3 and Table 1) except for the membrane resistance which has been reduced to
to mimic an expected high conductance in an in vivo state [21]. The synaptic time constant is set to
ms for the exponential synapse (Eq. 117) and
ms for the alpha-function synapse (Eq. 118).