Spatiotemporal Kernels Describing Hippocampal Nonlinear Dynamics
in Behaving Rats
T. P. Zanos1, S. H. Courellis1, R. Hampson2, V. Z. Marmarelis1, T. W. Berger1
Department of Biomedical Engineering, USC, Los Angeles, CA, zanos@usc.edu
2
Department of Physiology and Pharmacology, Wake Forest University, Winston-Salem, NC
1
Abstract:
Spatiotemporal nonlinear dynamic descriptors are introduced
to quantify hippocampal circuit dynamics in behaving rats.
The Volterra modeling approach is used to compute these
descriptors in the form of spatiotemporal kernels.
Electrophysiological data from several CA3 and CA1 cells
were recorded simultaneously using an array of penetrating
electrodes. This contemporaneous spike activity was used to
compute up to third order spatiotemporal kernels for the
multiple-input / single output case. Representative sets of
kernels illustrate the variability of the dynamics of the CA3CA1 functional mapping in space and time. A kernel
visualization approach is proposed to facilitate tracking
spatial and temporal changes of kernel dynamics.
Introduction:
The functional mapping between the CA3 and CA1 regions
of the hippocampus plays an important role in learning and
memory. Malfunction of this mapping due to aging or
damage could heavily impair cognitive function. Cortical
neuroprosthetics comprise a reasonable solution to restoring
such loss of functionality. However, a reliable quantitative
representation of this functional mapping is required before it
can be implemented in the neuroprosthetic. Such
representation ought to lead to a scalable, compact model
with predictive capabilities.
In previous studies, we used acute hippocampal slice
preparations to acquire data and create a nonparametric
model that quantified the CA3-CA1 functional mapping. It
was a single input / single output model, considering only
temporal nonlinear dynamics. Our modeling method was
based on the Volterra modeling approach adapted for Poisson
point-processes [1].
In this study, we use data from live, behaving rats recorded
contemporaneously from several spatially distinct sites at
CA3 and CA1. Consequently, the functional relationship
between the CA3 and the CA1 hippocampal region has a
spatial and a temporal dimension with nonlinear
characteristics.
Traditionally, investigators have employed parametric
methods to model this mapping, both in-vivo and in-vitro [2,
3]. Such methods lead to complex representations that may
not be suitable for the implementation of a neuroprosthetic
device in hardware. Thus, we employed the Volterra
modeling approach generalized in space and time. Our effort
focused on the computation of spatiotemporal nonlinear
dynamic descriptors in the form of spatiotemporal Volterra
kernels. We used natural neural activity recorded in
individual CA3 and CA1 neurons during a DNMS (DelayedNonMatch-to-Sample) task.
In this article, we introduce spatiotemporal kernels (up to
third order) of the CA3-CA1 functional mapping for specific
behavioral events during the DNMS tasks and we propose a
kernel visualization approach to facilitate their interpretation.
Methods:
A multi-electrode array of penetrating electrodes was used to
record the contemporaneous spike activity in the CA3 and
CA1 areas; a conceptual representation of it is shown in
Figure 1.
Figure 1: Conceptual representation of the multi-electrode array.
This array of electrodes recorded spike trains from multiple
cells in CA3 and CA1 of behaving rats, during the DNMS
task. The sequence of behavioral events during the DNMS
task included a Sample Event (rat hitting a lever), nose-pokes
(for distraction purposes) and a NonMatch Event (rat hitting a
lever different from the initial one). The recorded neural
activity was in the form of action potentials and converted to
binary spike sequences of variable interspike intervals.
This class of input / output datasets was used to compute the
spatiotemporal Volterra kernels of a third order model
mathematically formulated as follows:
y  n   k0  u1 ( n )  u2 ( n )  u3 ( n )
Q M 1
u1 ( n )    k1sq  m sq (n  m )
First - order Term
q 1 m  0
u2 ( n )   k2 sq sq  m1 , m2  sq1 (n  m1 ) sq2 (n  m2 ) Second - order Term
q1
q2
m1 m2
1
2
u3 ( n )   k3sq
q1
q2
q2
m1 m2
m3
1
sq s q
2
3
 m1 , m2 , m3  sq (n  m1 ) sq (n  m2 ) sq (n  m3 )
1
2
3
Third - order Term
where Q is the number of inputs sq(n), {k0, k1, k2, k3}
represent the zero, first, second, and third order Volterra
kernels , and y(n) denotes the output. The kernels were
computed using the Laguerre expansion method [4]. Using
the orthonormal set of Laguerre functions {Ll(m)} to expand
the kernels, we obtain:
L 1
k1 ( m )   c (1) l Ll ( m )
l 0
L 1 L 1
k2 (m1 , m2 )   c (2) l1l2 Ll1 (m1 ) Ll2 (m2 )
l1 0 l2 0
L 1 L 1 L 1
k3 (m1 , m2 , m3 )   c (3) l1l2l3 Ll1 (m1 ) Ll2 (m2 ) Ll3 (m3 )
l1 0 l2 0 l3 0
Βy using least-squares fitting, we estimate the expansion
coefficients and finally compute the kernels.
Results:
Spatiotemporal kernels were computed using data recorded
during the “Sample” behavioral event. Several instances of
the “Sample” behavioral event were considered across a
number of different DNMS trials. Figure 2 shows the first,
second, and third order spatio-temporal kernels for an array
of ten inputs in distinct spatial locations. A detailed view of
the kernels for one input is shown on Figure 3.
(A)
(B)
(C)
Figure 2: First (A), second (B) and third (C) order spatiotemporal nonlinear
dynamic descriptors.
Figure 3: Detailed view of kernels that characterize the mapping between
different cell types.
Discussion:
The computed spatiotemporal kernels reveal areas of
facilitatory and depressive behavior that vary as functions of
space and time. Inspection of Figure 2(A) suggests that the
first order kernels can start with a fast facilitatory or
depressive phase depending on the spatial location that is
followed by a slower facilitatory phase. Figure 2(B) shows
second order kernels that depending on the spatial location of
the input can be mostly facilitatory (e.g., kernels
corresponding to s1 and s2) or depressive (e.g., kernels
corresponding to s3 and s4). Similar interpretation is possible
for the third order kernels shown in Figure 2(C).
In our analysis, so far, we have not considered crossinteractions among the various inputs. Typically, crossinteractions are present in most neural systems and the
hippocampus is not an exception. We present this work as the
first step towards creating a rigorous framework that will
efficiently map nonlinear functional characteristics of neural
systems in both the temporal and the spatial domain.
References:
[1] G. Gholmieh, S.H. Courellis, V.Z. Marmarelis, T.W.
Berger (2002) An Efficient Method for Studying Short
Term Plasticity with Random Impulse Train Stimuli.
Journal of Neuroscience Methods (2002) 21(2), 111-127.
[2] W. B. Levy (2004) A Sequence Predicting CA3 IS a
Flexible Associator That Learns and Uses Context to
Solve Hippocampal-Like Tasks. Hippocampus (1996) 6,
579-590.
[3] E. D. Menschik, S. C. Yen, L. H. Finkel (1999) Modeland scale-independent performance of a hippocampal
CA3 network architecture. Neurocomputing (1999), 2627(1-3),443-453.
[4] V.Z. Marmarelis (1993) Identification of nonlinear
biological systems using Laguerre expansions of kernels
Annals of BiomedicalEngineering (1993) 21, 573-589.
Acknowledgments:
This work was supported by ERC(BMES), DARPA
(HAND), NIH (NIBIB).