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Local Field Potentials, Spikes and Modeling Strategies for Both Robert Haslinger Dept. of Brain and Cog. Sciences: MIT Martinos Center for Biomedical Imaging: MGH Robert Haslinger Dept. of Brain and Cog. Sciences: MIT Martinos Center for Biomedical Imaging: MGH

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Outline 1. Extra-cellular electric potentials, their origin, and their filtering 2. Local Field Potential and Continuous Models 3. Spikes and Generalized Linear Models 4. Example of GLM modeling in rat barrel cortex 1. Extra-cellular electric potentials, their origin, and their filtering 2. Local Field Potential and Continuous Models 3. Spikes and Generalized Linear Models 4. Example of GLM modeling in rat barrel cortex

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The Brain is a Complex Structured Network of Neurons

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Neural Activity Neurons have a ~ - 70 mV potential gradient across their membranes Neurons have a ~ - 70 mV potential gradient across their membranes Synaptic activity can depolarize the membrane Synaptic activity can depolarize the membrane Enough depolarization leads to a sharp (1msec 80 mV) change in potential (spike) which propagates down axons to other neurons Enough depolarization leads to a sharp (1msec 80 mV) change in potential (spike) which propagates down axons to other neurons synapses, leak currents, capacitive currents, spikes etc. all move charge across the neural membrane: GENERATE ELECTRIC POTENTIAL

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What do we actually measure ?

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Experiments record extra-cellular voltage changes Experiments record extra-cellular voltage changes Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes Experiments record extra-cellular voltage changes Experiments record extra-cellular voltage changes Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes

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What do we actually measure ? Experiments record extra-cellular voltage changes Experiments record extra-cellular voltage changes Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes Generally extra-cellular voltage is filtered into two types of signals: spikes and LFP Generally extra-cellular voltage is filtered into two types of signals: spikes and LFP Experiments record extra-cellular voltage changes Experiments record extra-cellular voltage changes Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes Generally extra-cellular voltage is filtered into two types of signals: spikes and LFP Generally extra-cellular voltage is filtered into two types of signals: spikes and LFP

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Local Field Potential

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Local Field Potential (LFP) Low pass filtered (0.1 - 250 Hz) signal Low pass filtered (0.1 - 250 Hz) signal “slower” processes, synapses, leak currents, capacitive currents etc. “slower” processes, synapses, leak currents, capacitive currents etc. Low pass filtered (0.1 - 250 Hz) signal Low pass filtered (0.1 - 250 Hz) signal “slower” processes, synapses, leak currents, capacitive currents etc. “slower” processes, synapses, leak currents, capacitive currents etc. Haslinger & Neuenschwander

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Local Field Potential (LFP) Low pass filtered (0.1 - 250 Hz) signal Low pass filtered (0.1 - 250 Hz) signal “slower” processes, synapses, leak currents, capacitive currents etc. “slower” processes, synapses, leak currents, capacitive currents etc. Haslinger & Devor

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Local Field Potential LFP generated through current “sinks” and “sources” LFP generated through current “sinks” and “sources”

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Local Field Potential LFP generated through current “sinks” and “sources” LFP generated through current “sinks” and “sources”

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Local Field Potential LFP generated through current “sinks” and “sources” LFP generated through current “sinks” and “sources”

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Local Field Potential LFP generated through current “sinks” and “sources” LFP generated through current “sinks” and “sources”

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Local Field Potential LFP generated through current “sinks” and “sources” LFP generated through current “sinks” and “sources” Can think of charge imbalances creating extracellular voltage Can think of charge imbalances creating extracellular voltage LFP generated through current “sinks” and “sources” LFP generated through current “sinks” and “sources” Can think of charge imbalances creating extracellular voltage Can think of charge imbalances creating extracellular voltage

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Local Field Potential LFP generated through current “sinks” and “sources” LFP generated through current “sinks” and “sources” Can think of charge imbalances creating extracellular voltage Can think of charge imbalances creating extracellular voltage Or can think in terms of voltage drops due to current loops, e.g. Ohm’s Law (V=IR) Or can think in terms of voltage drops due to current loops, e.g. Ohm’s Law (V=IR) LFP generated through current “sinks” and “sources” LFP generated through current “sinks” and “sources” Can think of charge imbalances creating extracellular voltage Can think of charge imbalances creating extracellular voltage Or can think in terms of voltage drops due to current loops, e.g. Ohm’s Law (V=IR) Or can think in terms of voltage drops due to current loops, e.g. Ohm’s Law (V=IR) current loop

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LFP results from the superposition of potentials from ALL sinks and sources We only see sinks and source pairs that don’t overlap with each other, or with sinks and sources from other neurons. We only see sinks and source pairs that don’t overlap with each other, or with sinks and sources from other neurons. Elongated pyramidal neurons, YES. Elongated pyramidal neurons, YES. Compact interneurons or layer IV stellate cells : NO Compact interneurons or layer IV stellate cells : NO Synchronous (across cells) events: YES Synchronous (across cells) events: YES Sinks not always excitatory, sources not always inhibitory Sinks not always excitatory, sources not always inhibitory We only see sinks and source pairs that don’t overlap with each other, or with sinks and sources from other neurons. We only see sinks and source pairs that don’t overlap with each other, or with sinks and sources from other neurons. Elongated pyramidal neurons, YES. Elongated pyramidal neurons, YES. Compact interneurons or layer IV stellate cells : NO Compact interneurons or layer IV stellate cells : NO Synchronous (across cells) events: YES Synchronous (across cells) events: YES Sinks not always excitatory, sources not always inhibitory Sinks not always excitatory, sources not always inhibitory

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LFP Changes With Position

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LFP Is Not Well Localized Spatially LFP localized within.5 - 3 mm (at best) LFP localized within.5 - 3 mm (at best) V~1/r … but in cortex its worse !!!! V~1/r … but in cortex its worse !!!! Caused by dendritic and cortical geometry Caused by dendritic and cortical geometry LFP localized within.5 - 3 mm (at best) LFP localized within.5 - 3 mm (at best) V~1/r … but in cortex its worse !!!! V~1/r … but in cortex its worse !!!! Caused by dendritic and cortical geometry Caused by dendritic and cortical geometry

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Pseudo 1D Geometry Cortex is a thin (2-3 mm thick) sheet Greatest anatomical variation perpendicular to the sheet Essentially a ONE DIMENSIONAL geometry In 1D, V ~ z, not 1/r

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LFP is Non-Local

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Characterizing LFP Highly complex continuous time signal Highly complex continuous time signal Intuition about phenomena at different time scales Intuition about phenomena at different time scales Can apply all sorts of signal processing techniques Can apply all sorts of signal processing techniques Helps to have some idea of what you’re looking for Helps to have some idea of what you’re looking for Highly complex continuous time signal Highly complex continuous time signal Intuition about phenomena at different time scales Intuition about phenomena at different time scales Can apply all sorts of signal processing techniques Can apply all sorts of signal processing techniques Helps to have some idea of what you’re looking for Helps to have some idea of what you’re looking for

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Signal Processing Techniques Fourier and Windowed Fourier Transform (multi- taper) (Chronux) Fourier and Windowed Fourier Transform (multi- taper) (Chronux) Wavelets and Multi-Resolution Analysis Wavelets and Multi-Resolution Analysis Time - Frequency representation Time - Frequency representation Hilbert Transform Hilbert Transform Power, Phase Power, Phase Empirical Mode Decomposition Empirical Mode Decomposition Autoregressive Modeling Autoregressive Modeling Coherency Analysis (Chronux) Coherency Analysis (Chronux) Granger Causality Granger Causality Information Theory Information Theory Fourier and Windowed Fourier Transform (multi- taper) (Chronux) Fourier and Windowed Fourier Transform (multi- taper) (Chronux) Wavelets and Multi-Resolution Analysis Wavelets and Multi-Resolution Analysis Time - Frequency representation Time - Frequency representation Hilbert Transform Hilbert Transform Power, Phase Power, Phase Empirical Mode Decomposition Empirical Mode Decomposition Autoregressive Modeling Autoregressive Modeling Coherency Analysis (Chronux) Coherency Analysis (Chronux) Granger Causality Granger Causality Information Theory Information Theory

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Frequency - ology Alpha (8-12 Hz) attention Alpha (8-12 Hz) attention Beta (12-20 Hz) Beta (12-20 Hz) Gamma (40-80 Hz) complex processing, mediated by inhibition Gamma (40-80 Hz) complex processing, mediated by inhibition Delta (1-4 Hz) slow wave sleep Delta (1-4 Hz) slow wave sleep Mu (8-12 Hz) but in motor cortex Mu (8-12 Hz) but in motor cortex Theta (4-8 Hz) Hippocampus Theta (4-8 Hz) Hippocampus Alpha (8-12 Hz) attention Alpha (8-12 Hz) attention Beta (12-20 Hz) Beta (12-20 Hz) Gamma (40-80 Hz) complex processing, mediated by inhibition Gamma (40-80 Hz) complex processing, mediated by inhibition Delta (1-4 Hz) slow wave sleep Delta (1-4 Hz) slow wave sleep Mu (8-12 Hz) but in motor cortex Mu (8-12 Hz) but in motor cortex Theta (4-8 Hz) Hippocampus Theta (4-8 Hz) Hippocampus

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Statistical Modeling of LFP System EEG, LFP covariates Linear Regression (Gaussian Model of Variability) Many standard methods for regression, model selection, goodness of fit and so forth

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Spikes

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Spikes : “high pass filtered” Extra-cellular voltage is high pass filtered and discrete spikes are identified through spike sorting Extra-cellular voltage is high pass filtered and discrete spikes are identified through spike sorting Neurons generating spikes are located near the electrode Neurons generating spikes are located near the electrode See spikes from all types of neurons (pyramids, interneurons etc.) See spikes from all types of neurons (pyramids, interneurons etc.) Functional distinctions based on spike shape (FS = inhibitory, RS = excitatory) Functional distinctions based on spike shape (FS = inhibitory, RS = excitatory) Neurons generating spikes are located near the electrode Neurons generating spikes are located near the electrode See spikes from all types of neurons (pyramids, interneurons etc.) See spikes from all types of neurons (pyramids, interneurons etc.) Functional distinctions based on spike shape (FS = inhibitory, RS = excitatory) Functional distinctions based on spike shape (FS = inhibitory, RS = excitatory)

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Spikes are discrete events Smooth into spike rate - continuous process Interspike interval distribution (ISI) Spectral techniques (multi-taper)

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Spikes are discrete events Smooth into spike rate - continuous process Interspike interval distribution (ISI) Spectral techniques (multi-taper) Point Process Statistical Modeling

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Introducing Generalized Linear Models System EEG, LFP covariates Linear Regression (Gaussian Model of Variability) System spikes covariates

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Conditional Intensity Function Spikes depend upon both external covariates (stimuli) and the previous history of the spiking process (t) = ( x(t) | H t ) (t) dt is the probability of a spike conditioned on the past spiking history H t and a function of the external covariates (stimuli) x(t)

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Conditional Intensity Function Spikes depend upon both external covariates (stimuli) and the previous history of the spiking process (t) = ( x(t) | H t ) (t) dt is the probability of a spike conditioned on the past spiking history H t and a function of the external covariates (stimuli) x(t) Our goal in statistical modeling is to get (t). Once we know that, we know “everything” (probability of any spike sequence for example)

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Regression for Event-Like Data “Standard” regression (linear or non-linear) assumes continuous data and Gaussian noise “Standard” regression (linear or non-linear) assumes continuous data and Gaussian noise Spikes are localized events, we should respect the nature of the data Spikes are localized events, we should respect the nature of the data A statistical model can be used for inference, prediction, decoding and simulation A statistical model can be used for inference, prediction, decoding and simulation There are standard techniques for modeling point process data, e.g. logistic regression and other Generalized Linear Models There are standard techniques for modeling point process data, e.g. logistic regression and other Generalized Linear Models “Standard” regression (linear or non-linear) assumes continuous data and Gaussian noise “Standard” regression (linear or non-linear) assumes continuous data and Gaussian noise Spikes are localized events, we should respect the nature of the data Spikes are localized events, we should respect the nature of the data A statistical model can be used for inference, prediction, decoding and simulation A statistical model can be used for inference, prediction, decoding and simulation There are standard techniques for modeling point process data, e.g. logistic regression and other Generalized Linear Models There are standard techniques for modeling point process data, e.g. logistic regression and other Generalized Linear Models

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Linear vs. Logistic Regression (t) = i i x i (t) restricted only by range of {x i }

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Linear vs. Logistic Regression (t) = i i x i (t) restricted only by range of {x i } log[ (t) / (1 - (t) ] = i i x i (t) is restricted between 0 and 1 is a PROBABILITY

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Linear vs. Logistic Regression (t) = i i x i (t) restricted only by range of {x i } log[ (t) / (1 - (t) ] = i i x i (t) is restricted between 0 and 1 is a PROBABILITY LINK FUNCTION

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Generalized Linear Models Logistic regression is one example of a Generalized Linear Model (GLM) Logistic regression is one example of a Generalized Linear Model (GLM) Can be solved by maximum likelihood estimation (log-concave problem) Can be solved by maximum likelihood estimation (log-concave problem) There exist efficient estimation techniques (iterative re-weighted least squares) There exist efficient estimation techniques (iterative re-weighted least squares) They can be solved in Matlab (glmfit.m) and almost all statistical packages They can be solved in Matlab (glmfit.m) and almost all statistical packages Logistic regression is one example of a Generalized Linear Model (GLM) Logistic regression is one example of a Generalized Linear Model (GLM) Can be solved by maximum likelihood estimation (log-concave problem) Can be solved by maximum likelihood estimation (log-concave problem) There exist efficient estimation techniques (iterative re-weighted least squares) There exist efficient estimation techniques (iterative re-weighted least squares) They can be solved in Matlab (glmfit.m) and almost all statistical packages They can be solved in Matlab (glmfit.m) and almost all statistical packages

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Possible Covariates to Include log[ (t) / (1 - (t)) ] = i i f i (stimulus)

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Possible Covariates to Include log[ (t) / (1 - (t)) ] = i i f i (stimulus) j j g j (spiking history)

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Possible Covariates to Include log[ (t) / (1 - (t)) ] = i i f i (stimulus) j j g j (spiking history) k k h k (ensemble spiking)

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Possible Covariates to Include log[ (t) / (1 - (t)) ] = i i f i (stimulus) j j g j (spiking history) k k h k (ensemble spiking) p p r p (LFP)

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Possible Covariates to Include log[ (t) / (1 - (t)) ] = i i f i (stimulus) j j g j (spiking history) k k h k (ensemble spiking) p p r p (LFP) Fitted parameters give the importance of different contributions

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Goodness-of-Fit Time Rescale Time-Rescaling Theorem: z i ’s are i.i.d. exponential rate 1 Kolmogorov-Smirnov (KS) Plot: ECDF(z i ) CDF(exp(1))

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GLM Example : Rat Barrel Cortex M. Andermann

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Inclusion of Different Covariates

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Time since stimulus i i B i (t) spline basis functions Inclusion of Different Covariates

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Time since stimulus Deflection Angle i i B i (t) spline basis functions cos ( 0 ) = 1 cos( ) - 2 sin( ) Inclusion of Different Covariates

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Time since stimulus Deflection Angle Spike History i i B i (t) spline basis functions cos ( 0 ) = 1 cos( ) - 2 sin( ) j j g j ( t - t j ) g(t) = 0 (no spike at t) = 1 (spike at t) Inclusion of Different Covariates

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log[ (t) / (1 - (t) ) ] = i i B i (t) + 1 cos( ) - 2 sin( ) j j g j ( t - t j ) FINAL MODEL

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log[ (t) / (1 - (t) ) ] = i i B i (t) + 1 cos( ) - 2 sin( ) j j g j ( t - t j ) FINAL MODEL History Term Often spike history effects account for most of the statistics !!!!!!!

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log[ (t) / (1 - (t) ) ] = i i B i (t) + 1 cos( ) - 2 sin( ) j j g j ( t - t j ) FINAL MODEL History Term Often spike history effects account for most of the statistics !!!!!!! refractory bursting

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Conclusions Important to understand the physical origins of what we record and model Important to understand the physical origins of what we record and model LFP and Spikes are fundamentally different types of data and require different modeling strategies LFP and Spikes are fundamentally different types of data and require different modeling strategies LFP requires some thought about what to model, but techniques are standard LFP requires some thought about what to model, but techniques are standard Spikes effectively described by probability but are point processes and require different techniques Spikes effectively described by probability but are point processes and require different techniques Logistic Regression (and other GLMs) for spikes. Kolmogorov Smirnov test for goodness of fit Logistic Regression (and other GLMs) for spikes. Kolmogorov Smirnov test for goodness of fit Rigorous model identification is important to determine the importance of different covariates. Rigorous model identification is important to determine the importance of different covariates. This can be a prelude to developing more effective BMIs This can be a prelude to developing more effective BMIs Important to understand the physical origins of what we record and model Important to understand the physical origins of what we record and model LFP and Spikes are fundamentally different types of data and require different modeling strategies LFP and Spikes are fundamentally different types of data and require different modeling strategies LFP requires some thought about what to model, but techniques are standard LFP requires some thought about what to model, but techniques are standard Spikes effectively described by probability but are point processes and require different techniques Spikes effectively described by probability but are point processes and require different techniques Logistic Regression (and other GLMs) for spikes. Kolmogorov Smirnov test for goodness of fit Logistic Regression (and other GLMs) for spikes. Kolmogorov Smirnov test for goodness of fit Rigorous model identification is important to determine the importance of different covariates. Rigorous model identification is important to determine the importance of different covariates. This can be a prelude to developing more effective BMIs This can be a prelude to developing more effective BMIs

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