NONLINEAR DYNAMICAL ANALYSIS OF FIRING PATTERNS OF GLOBUS PALLIDUS NEURONS Ramana Dodla and Charles J. Wilson

Department of Biology, University of Texas at San Antonio, 1 UTSA Circle, San Antonio, TX 78240

Motivation

1 ISIs do not follow gamma distribution 2 Log-normality

Main results

7 Structure in the ISI sequences

Acknowledgments

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06914c8, 5 min averages

<ISI>

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SD/sqrt(eσ2-1)

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1/<ISI> (Hz)

06914c8, 5 min averagesa=1.036, b=-0.006. rms of residuals = 0.005

y=1/x

least squares fit: a/<ISI>+b

1 hr long recordings of 14 cells. Each point is average of 5 min ISI train: 12 points per cell. Least squares fit: f(x)=a/<isi>+b. a=0.9239 b=-0.03

(RMS of residuals=0.06)

3 Serial and auto correlations

-0.4 -0.2 0 0.2 0.4 d (s)

STAC

Period=0.125 s (Freq=8 Hz) Rhythmicity is retained:

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�0.4

�0.2

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STA

C

06914 c8.d : 1 3 subsequences

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Seria

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oeff.

at l

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Cell #

14 cells, 5 min rec

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14 cells. Each recording 60 min

ISI density Survivor function Hazard function

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Skewness

Exponential

Gamma (a=2) (=2/sqrt(a))

Gamma (a=3)

Gamma (a=25)

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Kurtosis

(two points above 10)

Exponential

Gamma (a=2) (=3+6/a)

Gamma (a=3)

Gamma (a=25)

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CV

σ, SD of the log(ISIs)

Log-normal

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Skew

ness

σ, SD of the log(ISIs)

Log-normal

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Kurto

sis

σ, SD of the log(ISIs)

Log-normal

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ISI m

ean

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00) 0.25 s

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ISI mean (s) (n=100)

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Normalized variability, Xi

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Normalized variability, Xi

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Normalized variability, Xi

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Normalized variability, Xi

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Normalized variability, Xi

0 5 10 15 20 25 30 35 40 45 50Frequency (Hz)

Periodogram of normalized variability06914c8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6Normalized variability, Xi

SD=0.13

Address the basis of complex spike time patterns of globus pallidus in slice recordings of rat basal ganglia. Determine the nature of the interspike interval distributions Study the oscillatory mechanisms underlying the serial and spike time auto correlations. Find the underlying dimensions of the oscillatory mechanism.

The spike rate, variability, and the higher order moments are all found to be time dependent. The interspike interval distributions deviate significantly from Gaussian or gamma distributions. A multiplicative process like log-normal distribution is found to be match the statistics more closely than the other distributions. The pacemaking activity, however, is evident in the strong serial correlations and spike time autocorrelations.

Quantile-quantile plots: A match with the straight line indicates closeness to that distribution

Typical pacemaking neuron with refractory period

For a completely random process, this would be a constant.

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CV

Exponential

Gamma (a=2) (=1/sqrt(a))

Gamma (a=3)

Gamma (a=25)

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Den

sity

ISI (s)

<ISI>=0.08, SD=0.01, CV=0.12, 1/<ISI>=11.9 Hz

06o13c1

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Surv

ivor

func

tion

ISI (s)

06o13c1

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ard

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tion

* 0.0

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ISI (s)

06o13c1

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ISI q

uant

ile

Exponential quantile

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ISI q

uant

ile

Gamma (a=2) quantile

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ISI q

uant

ile

Log-normal quantile

06o13c1

Higher order moments reveal even stronger deviations of the ISI distributions from exponential, and gamma distributions

The coefficients of variation (CV) of the cells differ substantially from an exponential distribution. Gamma distribution with small shape factors failed to account for the variability.

Comparing data from 14 cells (5 min recordings) with higher order moments of exp and gamma dists. The red dots in the figures below correspond to the ISI data shown above

Comparing the CV, skewness, and kurtosis with the standard deviation of log intervals (σ)(See the next panel for "Log-normal" curves)

4 "Pause-triggered averages"

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Aver

age

ISI (

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Time BEFORE a pause of 0.3 s or more (s)

Pre-pause ISI averages

Cell slowing down well before a big pause

But the large pause is cuedby a slight speed up

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age

ISI (

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Time AFTER a pause of 0.3 s or more (s)

Post-pause ISI averages 06926c6 06926c6

Interspike intervals are considered to be part of a single connected curve, and such a curve is used to resample data at regular intervals. Pauses are examined by averaging segments of this curve similar to spike triggered averages, but now spike thresholds replaced by onset of pauses.

Strong serial correlations, lag=1 Non-circular contours:

Spike time autocorrelations

06926c6, 1hr

06926c6 All cells

All cells

Each 5 min

5 Correlation function

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C(r

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06926c6

ISIs can repeat over time, or any two pairs can occur in sufficient closeness over time.The number of such closely spaced ISI pairs (not necessarily the nearest neighbors) within a preferred distance (r) can reveal the underlying spatial dimension. This function is called the correlation function, C(r). C(r) is growing linearly for our ISI pairs as also expected for a random process suggesting that the cells are strongly influenced by noise dynamics.

6 Embedding dimension

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The underlying dimensions are revealed by time shifting the ISIs and working with pseudo-vectors: The vectors corresponding to three-dimensional embedding are shown below:

ISI1, ISI2, ISI3, ISI4, ISI5, ... ISI2, ISI3, ISI4, ISI5, ISI6, ... ISI3, ISI4, ISI5, ISI6, ISI7, ...

The embedding dimension is found to be at least 16 for this cell.

Fluctuations in the pacemaking activity of GP neurons Pacemaker activity and correlations of GP spike trains

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Statistics of 12 consecutive ISI segments. The adjusted standard deviation depends linearly on the average ISI, like that of a log-normal distribution.

5 minute averages (12 points per cell) of all 14 cells.

Moving ISI averages. A consecutive sequence of 100 ISIs are used in the averaging.

Fluctuations in the mean ISIs is captured by defining a normalized variability, the difference of the mean ISis around any given mean value normalized to that mean value: normalized variability, X(t).

The two variables defined above are used to form the coordinates of a phase plane. This dynamics of this system can now be studied.

The periodicity of this phase plane trajectory is displayed below visually by plotting the normalized variability vs. its time lagged coordinate.

Periodogram of the normalized variability (left), and the density histogram (right).

This work is supported by NIH grant No. NS047085

Conclusions Globus pallidus neurons fire spontaneously in slices with frequencies 3-30Hz.

Strong fluctuations of the ISIs are found that follow log-normal distributions. Gating kinetics could be responsible for the observed fluctuations.

σ

1s

τ