Local Field Potentials, Spikes and Modeling Strategies for Both

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

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

3. The Brain is a Complex Structured Network of Neurons

4. Neurons have a ~ - 70 mV potential gradient across their membranes 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 Neural Activity synapses, leak currents, capacitive currents, spikes etc. all move charge across the neural membrane: GENERATE ELECTRIC POTENTIAL

5. What do we actually measure ?

6. What do we actually measure ? • Experiments record extra-cellular voltage changes • Voltage changes generated by movement of charge (Na, K, Ca, Cl) across neuronal membranes

7. What do we actually measure ? • Experiments record extra-cellular voltage changes • 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

8. Local Field Potential

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

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

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

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

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

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

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

16. Local Field Potential • LFP generated through current “sinks” and “sources” • 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) current loop

17. 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. • Elongated pyramidal neurons, YES. • Compact interneurons or layer IV stellate cells : NO • Synchronous (across cells) events: YES • Sinks not always excitatory, sources not always inhibitory

18. LFP Changes With Position

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

20. 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

21. LFP is Non-Local

22. LFP is Non-Local

23. LFP is Non-Local

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

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

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

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

28. Spikes

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

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

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

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

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

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

35. Regression for Event-Like Data • “Standard” regression (linear or non-linear) assumes continuous data and Gaussian noise • Spikes are localized events, we should respect the nature of the data • 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

36. Linear vs. Logistic Regression (t) = ii xi(t) restricted only by range of {xi}

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

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

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

40. Possible Covariates to Include log[ (t) / (1 - (t)) ] = ii fi (stimulus)

41. Possible Covariates to Include log[ (t) / (1 - (t)) ] = ii fi (stimulus) jj gj (spiking history)

42. Possible Covariates to Include log[ (t) / (1 - (t)) ] = ii fi (stimulus) jj gj (spiking history) kk hk (ensemble spiking)

43. Possible Covariates to Include log[ (t) / (1 - (t)) ] = ii fi (stimulus) jj gj (spiking history) kk hk (ensemble spiking) pp rp (LFP)

44. Possible Covariates to Include log[ (t) / (1 - (t)) ] = ii fi (stimulus) jj gj (spiking history) kk hk (ensemble spiking) pp rp (LFP) Fitted parameters give the importance of different contributions

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

46. GLM Example : Rat Barrel Cortex M. Andermann

47. Inclusion of Different Covariates

48. Inclusion of Different Covariates ii Bi (t) Time since stimulus spline basis functions

49. Inclusion of Different Covariates ii Bi (t) Time since stimulus spline basis functions cos (0) = 1 cos() - 2 sin() Deflection Angle

50. Inclusion of Different Covariates ii Bi (t) Time since stimulus spline basis functions cos (0) = 1 cos() - 2 sin() Deflection Angle jj gj ( t - tj ) Spike History g(t) = 0 (no spike at t) = 1 (spike at t)