Frequency Dependence of Signal Power and Spatial Reach of the Local Field Potential

Despite its century-old use, the interpretation of local field potentials (LFPs), the low-frequency part of electrical signals recorded in the brain, is still debated. In cortex the LFP appears to mainly stem from transmembrane neuronal currents following synaptic input, and obvious questions regarding the ‘locality’ of the LFP are: What is the size of the signal-generating region, i.e., the spatial reach, around a recording contact? How far does the LFP signal extend outside a synaptically activated neuronal population? And how do the answers depend on the temporal frequency of the LFP signal? Experimental inquiries have given conflicting results, and we here pursue a modeling approach based on a well-established biophysical forward-modeling scheme incorporating detailed reconstructed neuronal morphologies in precise calculations of population LFPs including thousands of neurons. The two key factors determining the frequency dependence of LFP are the spatial decay of the single-neuron LFP contribution and the conversion of synaptic input correlations into correlations between single-neuron LFP contributions. Both factors are seen to give low-pass filtering of the LFP signal power. For uncorrelated input only the first factor is relevant, and here a modest reduction (<50%) in the spatial reach is observed for higher frequencies (>100 Hz) compared to the near-DC () value of about . Much larger frequency-dependent effects are seen when populations of pyramidal neurons receive correlated and spatially asymmetric inputs: the low-frequency () LFP power can here be an order of magnitude or more larger than at 60 Hz. Moreover, the low-frequency LFP components have larger spatial reach and extend further outside the active population than high-frequency components. Further, the spatial LFP profiles for such populations typically span the full vertical extent of the dendrites of neurons in the population. Our numerical findings are backed up by an intuitive simplified model for the generation of population LFP.     

Computing the Local Field Potential (LFP) from Integrate-and-Fire Network Models

Leaky integrate-and-fire (LIF) network models are commonly used to study how the spiking dynamics of neural networks changes with stimuli, tasks or dynamic network states. However, neurophysiological studies in vivo often rather measure the mass activity of neuronal microcircuits with the local field potential (LFP). Given that LFPs are generated by spatially separated currents across the neuronal membrane, they cannot be computed directly from quantities defined in models of point-like LIF neurons. Here, we explore the best approximation for predicting the LFP based on standard output from point-neuron LIF networks. To search for this best “LFP proxy”, we compared LFP predictions from candidate proxies based on LIF network output (e.g, firing rates, membrane potentials, synaptic currents) with “ground-truth” LFP obtained when the LIF network synaptic input currents were injected into an analogous three-dimensional (3D) network model of multi-compartmental neurons with realistic morphology, spatial distributions of somata and synapses. We found that a specific fixed linear combination of the LIF synaptic currents provided an accurate LFP proxy, accounting for most of the variance of the LFP time course observed in the 3D network for all recording locations. This proxy performed well over a broad set of conditions, including substantial variations of the neuronal morphologies. Our results provide a simple formula for estimating the time course of the LFP from LIF network simulations in cases where a single pyramidal population dominates the LFP generation, and thereby facilitate quantitative comparison between computational models and experimental LFP recordings in vivo.     

Estimation of population firing rates and current source densities from laminar electrode recordings

This model study investigates the validity of methods used to interpret linear (laminar) multielectrode recordings. In computer experiments extracellular potentials from a synaptically activated population of about 1,000 pyramidal neurons are calculated using biologically realistic compartmental neuron models combined with electrostatic forward modeling. The somas of the pyramidal neurons are located in a 0.4 mm high and wide columnar cylinder, mimicking a stimulus-evoked layer-5 population in a neocortical column. Current-source density (CSD) analysis of the low-frequency part (<500 Hz) of the calculated potentials (local field potentials, LFP) based on the ‘inverse’ CSD method is, in contrast to the ‘standard’ CSD method, seen to give excellent estimates of the true underlying CSD. The high-frequency part (>750 Hz) of the potentials (multi-unit activity, MUA) is found to scale approximately as the population firing rate to the power 3/4 and to give excellent estimates of the underlying population firing rate for trial-averaged data. The MUA signal is found to decay much more sharply outside the columnar populations than the LFP.     

Conversion of Phase Information into a Spike-Count Code by Bursting Neurons

Single neurons in the cerebral cortex are immersed in a fluctuating electric field, the local field potential (LFP), which mainly originates from synchronous synaptic input into the local neural neighborhood. As shown by recent studies in visual and auditory cortices, the angular phase of the LFP at the time of spike generation adds significant extra information about the external world, beyond the one contained in the firing rate alone. However, no biologically plausible mechanism has yet been suggested that allows downstream neurons to infer the phase of the LFP at the soma of their pre-synaptic afferents. Therefore, so far there is no evidence that the nervous system can process phase information. Here we study a model of a bursting pyramidal neuron, driven by a time-dependent stimulus. We show that the number of spikes per burst varies systematically with the phase of the fluctuating input at the time of burst onset. The mapping between input phase and number of spikes per burst is a robust response feature for a broad range of stimulus statistics. Our results suggest that cortical bursting neurons could play a crucial role in translating LFP phase information into an easily decodable spike count code.     

Disentanglement of local field potential sources by independent component analysis

The spontaneous activity of working neurons yields synaptic currents that mix up in the volume conductor. This activity is picked up by intracerebral recording electrodes as local field potentials (LFPs), but their separation into original informative sources is an unresolved problem. Assuming that synaptic currents have stationary placing we implemented independent component model for blind source separation of LFPs in the hippocampal CA1 region. After suppressing contaminating sources from adjacent regions we obtained three main local LFP generators. The specificity of the information contained in isolated generators is much higher than in raw potentials as revealed by stronger phase-spike correlation with local putative interneurons. The spatial distribution of the population synaptic input corresponding to each isolated generator was disclosed by current-source density analysis of spatial weights. The found generators match with axonal terminal fields from subtypes of local interneurons and associational fibers from nearby subfields. The found distributions of synaptic currents were employed in a computational model to reconstruct spontaneous LFPs. The phase-spike correlations of simulated units and LFPs show laminar dependency that reflects the nature and magnitude of the synaptic currents in the targeted pyramidal cells. We propose that each isolated generator captures the synaptic activity driven by a different neuron subpopulation. This offers experimentally justified model of local circuits creating extracellular potential, which involves distinct neuron subtypes.     

Somatostatin immunohistochemistry of hippocampal slices with lucifer yellow-stained pyramidal neurons responding to somatostatin.

We have combined electrophysiology and immunohistochemistry to study the somatostatin (SS) innervation of neurons in the rat hippocampal slice. After recording the intracellular response of a pyramidal CA1 neuron in vitro to SS, Lucifer Yellow was injected into the cell and the slice fixed and processed for immunohistochemical localization of SS in the vicinity of the recorded neuron. Most pyramidal neurons (70%) responded to SS with a hyperpolarization associated with marked slowing of spontaneous discharge and reduced input resistance. SS-containing elements either crossed, ran parallel or seemingly terminated on the Lucifer Yellow-filled SS-responsive cell. These occurrences of close proximity of apparent pre- and postsynaptic elements were observed in all layers of the CA1 region and may represent synaptic terminations of SS elements on a pyramidal neuron that are likely to elicit membrane hyperpolarizations.     

Estimating Extracellular Spike Waveforms from CA1 Pyramidal Cells with Multichannel Electrodes

Extracellular (EC) recordings of action potentials from the intact brain are embedded in background voltage fluctuations known as the “local field potential” (LFP). In order to use EC spike recordings for studying biophysical properties of neurons, the spike waveforms must be separated from the LFP. Linear low-pass and high-pass filters are usually insufficient to separate spike waveforms from LFP, because they have overlapping frequency bands. Broad-band recordings of LFP and spikes were obtained with a 16-channel laminar electrode array (silicone probe). We developed an algorithm whereby local LFP signals from spike-containing channel were modeled using locally weighted polynomial regression analysis of adjoining channels without spikes. The modeled LFP signal was subtracted from the recording to estimate the embedded spike waveforms. We tested the method both on defined spike waveforms added to LFP recordings, and on in vivo-recorded extracellular spikes from hippocampal CA1 pyramidal cells in anaesthetized mice. We show that the algorithm can correctly extract the spike waveforms embedded in the LFP. In contrast, traditional high-pass filters failed to recover correct spike shapes, albeit produceing smaller standard errors. We found that high-pass RC or 2-pole Butterworth filters with cut-off frequencies below 12.5 Hz, are required to retrieve waveforms comparable to our method. The method was also compared to spike-triggered averages of the broad-band signal, and yielded waveforms with smaller standard errors and less distortion before and after the spike.     

Pyramidal neuron as two-layer neural network

The pyramidal neuron is the principal cell type in the mammalian forebrain, but its function remains poorly understood. Using a detailed compartmental model of a hippocampal CA1 pyramidal cell, we recorded responses to complex stimuli consisting of dozens of high-frequency activated synapses distributed throughout the apical dendrites. We found the cell's firing rate could be predicted by a simple formula that maps the physical components of the cell onto those of an abstract two-layer "neural network." In the first layer, synaptic inputs drive independent sigmoidal subunits corresponding to the cell's several dozen long, thin terminal dendrites. The subunit outputs are then summed within the main trunk and cell body prior to final thresholding. We conclude that insofar as the neural code is mediated by average firing rate, a two-layer neural network may provide a useful abstraction for the computing function of the individual pyramidal neuron.     

Cortical Neurovascular Coupling Driven by Stimulation of Channelrhodopsin-2

While functional imaging is widely used in studies of the brain, how well the hemodynamic signal represents the underlying neural activity is still unclear. And there is a debate on whether hemodynamic signal is more tightly related to synaptic activity or action potentials. This study intends to address these questions by examining neurovascular coupling driven by pyramidal cells in the motor cortex of rats. Pyramidal cells in the motor cortex of rats were selectively transduced with the light sensitive cation channel channelrhodopsin-2 (ChR2). Electrophysiological recordings and optical intrinsic signal imaging were performed simultaneously and synchronously to capture the neural activity and hemodynamics induced by optical stimulation of ChR2-expressing pyramidal cells. Our results indicate that both synaptic activity (local field potential, LFP) and action potentials (multi-unit activity, MUA) are tightly related to hemodynamic signals. While LFPs in γ band are better in predicting hemodynamic signals elicited by short stimuli, MUA has better predictions to hemodynamic signals elicited by long stimuli. Our results also indicate that strong nonlinearity exists in neurovascular coupling.     

Gain modulation from background synaptic input

Gain modulation is a prominent feature of neuronal activity recorded in behaving animals, but the mechanism by which it occurs is unknown. By introducing a barrage of excitatory and inhibitory synaptic conductances that mimics conditions encountered in vivo into pyramidal neurons in slices of rat somatosensory cortex, we show that the gain of a neuronal response to excitatory drive can be modulated by varying the level of "background" synaptic input. Simultaneously increasing both excitatory and inhibitory background firing rates in a balanced manner results in a divisive gain modulation of the neuronal response without appreciable signal-independent increases in firing rate or spike-train variability. These results suggest that, within active cortical circuits, the overall level of synaptic input to a neuron acts as a gain control signal that modulates responsiveness to excitatory drive.     

Identifying and Tracking Simulated Synaptic Inputs from Neuronal Firing: Insights from In Vitro Experiments

Accurately describing synaptic interactions between neurons and how interactions change over time are key challenges for systems neuroscience. Although intracellular electrophysiology is a powerful tool for studying synaptic integration and plasticity, it is limited by the small number of neurons that can be recorded simultaneously in vitro and by the technical difficulty of intracellular recording in vivo. One way around these difficulties may be to use large-scale extracellular recording of spike trains and apply statistical methods to model and infer functional connections between neurons. These techniques have the potential to reveal large-scale connectivity structure based on the spike timing alone. However, the interpretation of functional connectivity is often approximate, since only a small fraction of presynaptic inputs are typically observed. Here we use in vitro current injection in layer 2/3 pyramidal neurons to validate methods for inferring functional connectivity in a setting where input to the neuron is controlled. In experiments with partially-defined input, we inject a single simulated input with known amplitude on a background of fluctuating noise. In a fully-defined input paradigm, we then control the synaptic weights and timing of many simulated presynaptic neurons. By analyzing the firing of neurons in response to these artificial inputs, we ask 1) How does functional connectivity inferred from spikes relate to simulated synaptic input? and 2) What are the limitations of connectivity inference? We find that individual current-based synaptic inputs are detectable over a broad range of amplitudes and conditions. Detectability depends on input amplitude and output firing rate, and excitatory inputs are detected more readily than inhibitory. Moreover, as we model increasing numbers of presynaptic inputs, we are able to estimate connection strengths more accurately and detect the presence of connections more quickly. These results illustrate the possibilities and outline the limits of inferring synaptic input from spikes.     

Instantaneous Non-Linear Processing by Pulse-Coupled Threshold Units

Contemporary theory of spiking neuronal networks is based on the linear response of the integrate-and-fire neuron model derived in the diffusion limit. We find that for non-zero synaptic weights, the response to transient inputs differs qualitatively from this approximation. The response is instantaneous rather than exhibiting low-pass characteristics, non-linearly dependent on the input amplitude, asymmetric for excitation and inhibition, and is promoted by a characteristic level of synaptic background noise. We show that at threshold the probability density of the potential drops to zero within the range of one synaptic weight and explain how this shapes the response. The novel mechanism is exhibited on the network level and is a generic property of pulse-coupled networks of threshold units.     

A nonlinear cascade model for action potential encoding in an insect sensory neuron.

Action potential encoding in the cockroach tactile spine neuron can be represented as a single-input single-output nonlinear dynamic process. We have used a new functional expansion method to characterize the nonlinear behavior of the neural encoder. This method, which yields similar kernels to the Wiener method, is more accurate than the latter and is efficient enough to obtain reasonable kernels in less than 15 min using a personal computer. The input stimulus was band-limited white Gaussian noise and the output consisted of the resulting train of action potentials, which were unitized to give binary values. The kernels and the system input-output signals were used to identify a model for encoding comprising a cascade of dynamic linear, static nonlinear, and dynamic linear components. The two dynamic linear components had repeatable and distinctive forms with the first being low-pass and the second being high-pass. The static nonlinearity was fitted with a fifth-order polynomial function over several input amplitude ranges and had the form of a half-wave rectifier. The complete model gave a good approximation to the output of the neuron when both were subjected to the same novel white noise input signal.     

Modeling the spatial reach of the LFP

The local field potential (LFP) reflects activity of many?neurons in the vicinity of the recording electrode and is therefore useful for studying local network dynamics. Much of the nature of the LFP is, however, still unknown. There are, for instance, contradicting reports on the spatial extent of the region generating the LFP. Here, we use a detailed biophysical modeling approach to investigate the size of the contributing region by simulating the LFP from a large number of neurons around the electrode. We find that the size of the generating region depends on the neuron morphology, the synapse distribution, and the correlation in synaptic activity. For uncorrelated activity, the LFP represents cells in a small region (within a radius of a few hundred micrometers). If the LFP contributions from different cells are correlated, the size of the generating region is determined by the spatial extent of the correlated activity.     

Extracting non-linear integrate-and-fire models from experimental data using dynamic I–V curves

The dynamic I–V curve method was recently introduced for the efficient experimental generation of reduced neuron models. The method extracts the response properties of a neuron while it is subject to a naturalistic stimulus that mimics in vivo-like fluctuating synaptic drive. The resulting history-dependent, transmembrane current is then projected onto a one-dimensional current–voltage relation that provides the basis for a tractable non-linear integrate-and-fire model. An attractive feature of the method is that it can be used in spike-triggered mode to quantify the distinct patterns of post-spike refractoriness seen in different classes of cortical neuron. The method is first illustrated using a conductance-based model and is then applied experimentally to generate reduced models of cortical layer-5 pyramidal cells and interneurons, in injected-current and injected- conductance protocols. The resulting low-dimensional neuron models—of the refractory exponential integrate-and-fire type—provide highly accurate predictions for spike-times. The method therefore provides a useful tool for the construction of tractable models and rapid experimental classification of cortical neurons.     

Role of microstructure of nerve impulse train in relation to transmission of neuronal activity and coding mechanism of neural information

Input-output relation were of giant neurons of a marine mollusc, Onchidium verruculatum, and a computer-simulated neuron investigated in terms of microstructure of nerve impulse train. The microstructure of input impulse train, the size of a unitary EPSP, and the extent of spontaneous firing activity of a single neuron had an important influence upon the effective summation of arriving synaptic inputs, the elicitation of output spikes, and intervals between succeeding output spikes. The neuron responded differently to respective input trains with different time structures, i.e. it discriminated input time pattern to various degrees. The manner in discrimination of input time pattern was dependent on the size of the unitary EPSP and the extent of the spontaneous firing activity, if it had. Some discussions were made with regard to possible coding systems of neural signal, assuming a frequency code and/or a pattern code.     

The neural basis of the blood-oxygen-level-dependent functional magnetic resonance imaging signal

Magnetic resonance imaging (MRI) has rapidly become an important tool in clinical medicine and biological research. Its functional variant (functional magnetic resonance imaging; fMRI) is currently the most widely used method for brain mapping and studying the neural basis of human cognition. While the method is widespread, there is insufficient knowledge of the physiological basis of the fMRI signal to interpret the data confidently with respect to neural activity. This paper reviews the basic principles of MRI and fMRI, and subsequently discusses in some detail the relationship between the blood-oxygen-level-dependent (BOLD) fMRI signal and the neural activity elicited during sensory stimulation. To examine this relationship, we conducted the first simultaneous intracortical recordings of neural signals and BOLD responses. Depending on the temporal characteristics of the stimulus, a moderate to strong correlation was found between the neural activity measured with microelectrodes and the BOLD signal averaged over a small area around the microelectrode tips. However, the BOLD signal had significantly higher variability than the neural activity, indicating that human fMRI combined with traditional statistical methods underestimates the reliability of the neuronal activity. To understand the relative contribution of several types of neuronal signals to the haemodynamic response, we compared local field potentials (LFPs), single- and multi-unit activity (MUA) with high spatio-temporal fMRI responses recorded simultaneously in monkey visual cortex. At recording sites characterized by transient responses, only the LFP signal was significantly correlated with the haemodynamic response. Furthermore, the LFPs had the largest magnitude signal and linear systems analysis showed that the LFPs were better than the MUAs at predicting the fMRI responses. These findings, together with an analysis of the neural signals, indicate that the BOLD signal primarily measures the input and processing of neuronal information within a region and not the output signal transmitted to other brain regions.     

Theoretical Analysis of the Local Field Potential in Deep Brain Stimulation Applications

Deep brain stimulation (DBS) is a common therapy for treating movement disorders, such as Parkinson’s disease (PD), and provides a unique opportunity to study the neural activity of various subcortical structures in human patients. Local field potential (LFP) recordings are often performed with either intraoperative microelectrodes or DBS leads and reflect oscillatory activity within nuclei of the basal ganglia. These LFP recordings have numerous clinical implications and might someday be used to optimize DBS outcomes in closed-loop systems. However, the origin of the recorded LFP is poorly understood. Therefore, the goal of this study was to theoretically analyze LFP recordings within the context of clinical DBS applications. This goal was achieved with a detailed recording model of beta oscillations (∼20 Hz) in the subthalamic nucleus. The recording model consisted of finite element models of intraoperative microelectrodes and DBS macroelectrodes implanted in the brain along with multi-compartment cable models of STN projection neurons. Model analysis permitted systematic investigation into a number of variables that can affect the composition of the recorded LFP (e.g. electrode size, electrode impedance, recording configuration, and filtering effects of the brain, electrode-electrolyte interface, and recording electronics). The results of the study suggest that the spatial reach of the LFP can extend several millimeters. Model analysis also showed that variables such as electrode geometry and recording configuration can have a significant effect on LFP amplitude and spatial reach, while the effects of other variables, such as electrode impedance, are often negligible. The results of this study provide insight into the origin of the LFP and identify variables that need to be considered when analyzing LFP recordings in clinical DBS applications.     

Pre-processing and transfer entropy measures in motor neurons controlling limb movements

Directed information transfer measures are increasingly being employed in modeling neural system behavior due to their model-free approach, applicability to nonlinear and stochastic signals, and the potential to integrate repetitions of an experiment. Intracellular physiological recordings of graded synaptic potentials provide a number of additional challenges compared to spike signals due to non-stationary behaviour generated through extrinsic processes. We therefore propose a method to overcome this difficulty by using a preprocessing step based on Singular Spectrum Analysis (SSA) to remove nonlinear trends and discontinuities. We apply the method to intracellular recordings of synaptic responses of identified motor neurons evoked by stimulation of a proprioceptor that monitors limb position in leg of the desert locust. We then apply normalized delayed transfer entropy measures to neural responses evoked by displacements of the proprioceptor, the femoral chordotonal organ, that contains sensory neurones that monitor movements about the femoral-tibial joint. We then determine the consistency of responses within an individual recording of an identified motor neuron in a single animal, between repetitions of the same experiment in an identified motor neurons in the same animal and in repetitions of the same experiment from the same identified motor neuron in different animals. We found that delayed transfer entropy measures were consistent for a given identified neuron within and between animals and that they predict neural connectivity for the fast extensor tibiae motor neuron.