Measuring instantaneous frequency of local field potential oscillations using the Kalman smoother
Introduction
Understanding how local field potentials (LFPs) are modulated in amplitude and frequency may facilitate the use of LFP activity to infer neural function. To this end, determining whether or not small timescale variations in LFP activity correspond to functional or noise processes is important. In particular, the instantaneous oscillation frequency (iFreq) and frequency modulation (FM) of brain rhythms may be informative measures of network state. Here, we derive and discuss a novel analytical framework that will facilitate the characterization of LFP activity using iFreq and FM.
The activation of neural circuits and the characteristics of LFP activity are intimately tied to cognitive function and behavioral state. For example, during sleep, slow oscillations (<1 Hz) observed in the neocortex emerge when thalamocortical circuits are engaged (Steriade et al., 1993); in addition, delta waves (1–4 Hz) and spindles (7–14 Hz) are found to be superimposed on the positive half-wave of such slow oscillations (Steriade, 2006). In the hippocampus (HPC), the prominent θ-rhythm (6–12 Hz) is associated with sensorimotor integration during locomotion and can be sufficiently driven by cholinergic activity from the medial septal nuclei and diagonal band of Broca (Bland, 1986). The θ-rhythm has also been linked to cognitive function; Jones et al. demonstrated that the θ-rhythm in the prefrontal cortex and hippocampus became more coherent during behavioral epochs where more demand is placed on working memory (Jones and Wilson, 2005). Communication between the hippocampus and prefrontal cortex may also be an important component of the memory consolidation circuit; Molle et al. found that the phase of the slow oscillation and the amount of multi-unit activity in the prefrontal cortex was strongly correlated with the activation of hippocampal sharp-wave ripple events (Molle et al., 2006). Coherent gamma band activity (20–70 Hz) between brain regions have been shown to coordinate activity between brain structures during learning (Miltner et al., 1999, Popescu et al., 2009). The ability to characterize LFP activity with both temporal and spectral precision may help us to understand the neural computations underlying brain function.
The spectrogram, which is commonly used to quantify and visualize LFP characteristics, may be computed using two families of algorithms: parametric and non-parametric. The non-parametric class of algorithms generally require very few assumptions to be made about the data, but have well-known trade-offs between temporal resolution, spectral resolution, and estimation uncertainty (Percival and Walden, 1993). Because of these trade-offs, it has been difficult to systematically explore rhythmic field potential activity on the same timescale of neuronal interactions. Alternatively, parametric methods involve the use of an explicit data-generating model. More specifically, an autoregressive (AR) process may be used to model activity that is generated by one or more superimposed and noisy sinusoids, like the activity found in LFP recordings. The AR model is composed of parameters, which define the general structure of the model, and coefficients, which are realized by fitting the AR model to the data (Percival and Walden, 1993). Although visualization of the power spectral density (PSD) is a powerful inference tool, the primary advantage of the AR model is that the underlying process that produced the observed data can be inferred directly from the AR parameters and coefficients without ever constructing a PSD plot or spectrogram. Therefore, it is advantageous to estimate quantities such as iFreq and FM from LFP data using AR modeling.
In this paper, we are most interested in a group of solutions that fall under the class of “time-varying autoregressive models” or TVARs (Arnold et al., 1998, Bartoli and Cerutti, 1983, Baselli et al., 1997, Bohlin, 1977, Foffani et al., 2004, Tarvainen et al., 2004, Zetterberg, 1969). More specifically, TVARs explicitly model LFP activity as being generated by one or more superimposed, stochastic, dynamic oscillators. In general, TVARs largely consist of a three-step process: (1) assuming a general structure for the AR model, (2) employing an adaptive filter framework for dynamic AR model estimation, and (3) using the resulting TVAR model to infer oscillatory dynamics of the data. Although considerable progress has been made in these areas, with respect to LFP analysis, there are non-trivial obstacles to overcome before we can achieve the goal of performing reliable estimation of iFreq and FM from single-trial data.
In particular, the algorithm we developed addresses the following significant challenges. First, the combination of brain rhythms that are activated, and their relative amplitude of activation, is truly dynamic over small time-intervals; therefore, an important question is how to choose the AR model parameters so that the resulting TVAR model characterizes the data correctly during times of interest. Second, brain rhythms are characterized by activity occurring in well established frequency bands, however, no elegant methodology exists for constraining the peaks of the AR-PSD to pre-defined frequency bands of interest; for example, when there are zero or multiple peaks within a frequency band, the interpretation of the model may be ambiguous. Third, the choice of adaptive filter framework can greatly affect the quality of the TVAR model estimates. Last, a challenge is to understand the assumptions of the adaptive filter, and to ensure that the adaptive filter and data input are mutually compatible.
Hereafter, we derive and apply a fixed-interval Kalman smoother based TVAR model to track the iFreq and compute the FM of oscillations present in real and simulated LFP data. A Kalman filter based approach was chosen over other adaptive filters for the wealth of existing theoretical and practical knowledge from which to draw from (Arnold et al., 1998, Mendel, 1987, Tarvainen et al., 2004), and more specifically, for its ability to reduce estimation delays that are known to occur with adaptive filtering (Tarvainen et al., 2004). During the process, we illustrate common situations that would lead to the erroneous estimation of iFreq and FM. We discuss and provide examples of when these situations may arise and provide concrete methods for detecting and side-stepping these conditions. The final algorithm was independently validated by comparing our AD-KS algorithm to Hilbert transform and short-time Fourier transform algorithms. Finally, application of the AD-KS was demonstrated by quantifying the frequency structure of brief ripple oscillations recorded from the CA1 subfield of the rat hippocampus.
Section snippets
Methods
We begin the methodology section by defining a general model for a single stochastic and dynamic oscillation that is appropriate for characterizing rhythmic field potential activity. The narrow-band oscillatory model is fully defined by two functions of time: instantaneous frequency (iFreq) and instantaneous amplitude (iAMP). Given a brain rhythm of interest, the iAMP of the rhythm may be functionally related to behavioral state (Sinnamon, 2006, Wyble et al., 2004). Unlike the iAMP measure,
Results
Information transfer between the hippocampus and neocortex is important for the consolidation of spatial and episodic memory. This process of information transfer is referred to as memory consolidation and may be mediated by a phenomena called “ensemble sequence replay” (Foster and Wilson, 2006, Lee and Wilson, 2002). We know that this process of replay is associated with a rise in multi-unit activity and the presence of ripples (100–250 Hz oscillations with marked increases in power lasting, on
Discussion
We developed a compact Kalman smoother algorithm for estimating iFreq with minimized bias. Our method was designed to be simple and interpretable, so that it may be more accessible to the neuroscience community. Furthermore, the interpretation of the iFreq and FM measures is facilitated by the fact that brain rhythms are generally band-limited phenomena, and that our measurements are restricted to physiologically relevant frequency bands.
We illustrated several characteristics of the general
Acknowledgements
This work was funded in part by the following grants: NIH/NIMH R01 MH59733, NIH/NIHLB R01 HL084502, and the Singleton Graduate Student Fellowship. We thank members of the Wilson Lab, Brown Lab, and Tonegawa Lab for helpful comments.
References (47)
- et al.
Instantaneous modulation of gamma oscillation frequency by balancing excitation with inhibition
Neuron
(2009) - et al.
An optimal linear filter for the reduction of noise superimposed to the EEG signal
J Biomed Eng
(1983) - et al.
Tracking of time-varying frequency of sinusoidal signals
Signal Process
(1999) The physiology and pharmacology of hippocampal-formation theta rhythms
Prog Neurobiol
(1986)Analysis of EEG signals with changing spectra using a short-word Kalman estimator
Math Biosci
(1977)- et al.
Temporal structure in spatially organized neuronal ensembles: a role for interneuronal networks
Curr Opin Neurobiol
(1995) - et al.
Reduced spike-timing reliability correlates with the emergence of fast ripples in the rat epileptic hippocampus
Neuron
(2007) - et al.
Characterizing in vitro hippocampal ripples using time-frequency analysis
Neurocomputing
(2005) - et al.
Memory of sequential experience in the hippocampus during slow wave sleep
Neuron
(2002) Decline in hippocampal theta activity during cessation of locomotor approach sequences: amplitude leads frequency and relates to instrumental behavior
Neuroscience
(2006)
Grouping of brain rhythms in corticothalamic systems
Neuroscience
Hippocampal electrical activity and voluntary movement in rat
Electroencephalogr Clin Neurophysiol
Modelling non-stationary variance in EEG time series by state space GARCH model
Comput Biol Med
Estimation of parameters for a linear difference equation with application to EEG analysis
Math Biosci
Using a spatio-temporal dynamic state-space model with the EM algorithm to patch gaps in daily riverflow series
Hydrol Earth Syst Sci
Linear optimal control
Adaptive AR modeling of nonstationary time series by means of Kalman filtering
IEEE Trans Biomed Eng
Spectral decomposition in multichannel recordings based on multivariate parametric identification
IEEE Trans Biomed Eng
Induction of sharp wave-ripple complexes in vitro and reorganization of hippocampal networks
Nat Neurosci
Time series analysis: forecasting and control
Rhythms of the brain
High-frequency oscillations in the output networks of the hippocampal-entorhinal axis of the freely behaving rat
J Neurosci
Physiological patterns in the hippocampo-entorhinal cortex system
Hippocampus
Cited by (26)
Gearbox Fault Detection Using Synchro-squeezing Transform
2016, Procedia EngineeringModeling multiscale causal interactions between spiking and field potential signals during behavior
2022, Journal of Neural EngineeringInvestigation of vibration data-based human load monitoring system
2021, Structural Health MonitoringIdentification of system transfer function of a ship mounted two-dof manipulator under ocean random waves
2021, Proceedings of the International Offshore and Polar Engineering ConferenceA Computationally-Efficient, Online-Learning Algorithm for Detecting High-Voltage Spindles in the Parkinsonian Rats
2020, Annals of Biomedical Engineering
- 1
Address: Picower Institute for Learning and Memory, Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, 43 Vassar Street, Building 46, Room 5233, Cambridge, MA 02139, United States. Tel.: +1 617 253 2046.
- 2
Address: Department of Brain and Cognitive Sciences, Division of Health Sciences and Technology, Massachusetts Institute of Technology, 43 Vassar Street, Building 46, Room 6079, Cambridge, MA 02139, United States. Tel.: +1 617 324 1880.
- 3
Address: Harvard Medical School, 55 Fruit Street, GRJ 4, Boston, MA 02114-2696, United States. Tel.: +1 617 724 1061.