Transcript RBorisyuk-Hoppensteadt-APR-2008

Oscillatory Models of
Hippocampal Activity and
Memory
Roman Borisyuk
University of Plymouth, UK
In collaboration with
Frank Hoppensteadt
New York University
Outline
• Oscillatory model of Hippocampal Activity
• Memorization of sequences of events
• Theory of epineuronal memory
Publications
1. Borisyuk R.M. and Hoppensteadt, F. (1998) Memorizing and recalling
spatial-temporal patterns in an oscillator model of the hippocampus.
Biosystems, v.48, 3-10.
2. Borisyuk R., Denham M., Denham S. and Hoppensteadt F. (1999)
Computational models of predictive and memory-related functions of
the hippocampus. Reviews in the Neurosciences, v.10, pp.213-232.
3. Borisyuk R., Hoppensteadt F. (1999) Oscillatory model of the
hippocampus: A study of spatio-temporal patterns of neural activity.
Biological Cybernetics, v. 81, no.4, pp 359-371.
4. Borisyuk R., Denham M., Kazanovich Y., Hoppensteadt F.
Vinogradova O. (2000). An Oscillatory Neural Network Model of
Sparse Distributed Memory and Novelty Detection. BioSystems,
58:265-272
5. Borisyuk R., Denham M., Kazanovich Y., Hoppensteadt F.,
Vinogradova O., (2001). Oscillatory Model of Novelty Detection.
Network: Computation in Neural System, 12: 1-20
6. Borisyuk R. and Hoppensteadt F. (2004) A theory of epineuronal
memory. Neural Networks, 17:1427-1436.
Chain Model of Spatio-Temporal Activity
We model activity of the hippocampus by a chain of interactive
oscillators corresponding to lamellas.
Each oscillator has two theta modulated inputs with time shift
which controls resulting activity pattern (hippocampal bar
code).
System demonstrates a wide variety of dynamics:
synchronization, non-linear resonance, chaotic activity, etc.
Septum
V(t+1 )
S1
V(t+N )
...
V(t+1 )
Entorhinal Cortex
SN
V(t+N )
Borisyuk &
Hoppensteadt, 1999,
Biological
Cybernetics
Model Description
We study this model analytically using VCONs (Hoppeansteadt, 1975)
and computationally using W-C oscillator (Wilson & Cowan, 1972).
En(t) and In(t) are average activities of
excitatory and inhibitory populations;
Z() is sigmoid;
Rn and Vn describe interactions with
neighbours;
Pn and Qn are periodic inputs
D=C –S controls patterns of activity
Gamma and Theta Rhythms of Single Oscillator
Single oscillator under influences
of two inputs can demonstrate
complex behaviour with
slow (theta) and fast (gamma)
components
0
200
400 t
Recoding from hippocampal population
(Van Quyen & Bragin, 2007)
Spatio-Temporal Patterns Hippocampal Bar Code
D=5
TIME
Input:
SEPTAL HIPPOCAMPUS
sin( 2ft) Phase deviation D is a key
parameter which coltrols
dynamics of hippocampal
activity
TIME
D=18
Input:
EC
sin( 2ft  D )
Borisyuk & Hoppensteadt,
1999, Biological Cybernetics
Phase/Frequency Coding and Novelty Detection
• Equations of ONN dynamics:
q
n
d kj
v
w
= 2kj   sin( 20t   ij   kj )   g1 (alj ) sin(  l j   kj )
dt
n i =1
q l =1
n
da
1

j
2
j 
=  ak  g 2   cos  ( 0 t   ij   k ) 
dt
 n i =1

j
k
j

d kj
d

j
j
k 

=  g1 (ak )   k 
dt
dt 

exp(( x   i ) / i )
gi ( x) =
, (i = 1,2)
1  exp(( x   i ) / i )
Borisyuk, Denham, Kazanovich,
Hoppensteadt, Vinogradova
(2000,2001)
Model Description
Natural
9
frequencies
Dynamics of
oscillator’
frequencies is
governed by
the learning
rule: here we
do not modify
connection
strengths,
instead we
adjust natural
frequency
8.5
8
G3
7.5
7
G2
6.5
6
5.5
5
G1
4.5
4
time
0


=5
500


1000


=7


1500

2000


=8
2500

Stimulation
Dynamics of Frequencies and Amplitudes
Resonant state
Non-Resonant state
7,6
7,6
7,4
Natural frequencies
7,2
7
6,8
6,6
7,2
7
6,8
6,6
6,4
6,4
0
1
2
3
4
5
6
7
8
9
10
11
12 13
0
14
1
2
3
4
5
6
7
8
9
10
11
12 13
14
Time
Time
Amplitude
Natural frequencies
7,4
#
Time
Novelty Detection: Sparse Coding
Example of sparse coding: 10 object
are coded by 2000 groups
The bar’s height is
proportional to the
number of resonant
oscillators in the group.
O
H
E
L2
L1
W
O
D
R
L
0
2000
The arrow indicates
coincidence of resonant
oscillator groups for the
same symbols “O”
Oscillatory Memory of Sequences
The learning rule is temporally asymmetric, and it
takes into account the activity level of pre- and
post-”synaptic” neurons in two contiguous time
windows.
Recall by the network is fast: All memorized
patterns of sequences are reproduced in the
correct order during the same time window with
a short delay.
Borisyuk, Denham, Denham, Hoppensteadt (1999)
Asymmetric Learning Rule (analog of STDP)
Borisyuk Denham,
Denham, Hoppensteadt,
1999, Rev in Neurosc.
w (m  2) = w (m  1) 
n, j
n, j
(   )   ( E
j
max
(m)  h)  ( E
n
max
j
w n,j
(m  1)  h)  
n
Activity
Threshold h
Tm
Tm+1
Time
Oscillatory Memory
0.2
0.4
time
0.6
Example of ONN dynamics. Oscillator consists of 10
excitatory (RED) and 10 inhibitory (BLUE) integrate and fire
units with all-to-all connections. The background activity is
low. The external input is applied to some group of oscillators
during time window. Three time windows are shown.
ONN Memory: Sequence of 5 Patterns
t
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
ONN Memory: Two Sequences
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
t
Reverse Replay (Wilson Lab, MIT)
Place cell 1 fires
Place cell 2 fires
Place cell 3 fires
Reverse replay
Foster & Wilson, Nature, 2006
Reverse Replay with Anti STDP
Reverse Replay with Anti STDP
Forward and Reverse Replay
A series of neuronal place-fields,
which, when ordered according to
the peak in-field firing rates,
comprise the place-field sequence
“template”.
Each neuron’s place-field is shown
in a different color.
Some sample forward and reverse
correlated events from these
neurons (same coloring) during
immobility.
Forward replay
Diba & Buzsaki,
Nature Neurosc 2007
Preplay and Replay
Forward preplay
Diba & Buzsaki,
Nature Neurosc 2007
Reverse replay
Spike trains of 13 neurons before, during, and after a single lap (CA1 local field
potential shown on top; velocity of the rat shown in the lower panel). The left and right
insets magnify 250-ms sections of the spike train, depicting forward preplay and reverse
replay, respectively.
Epineuronal Memory
A theory of epineuronal memory includes a
hierarchical structure of variables and
parameters that allows us to consider learning
and memory processes as being on a variable
landscape that is sculptured by reward signals.
During fast dynamics, the landscape is attractive
quasi-static surface that then slowly guides the
system into basin of attraction of the metastable
state.
A novel mathematical model of Epineuronal
Memory is developed that is based on a
temporally evolving mnemonic function M, which
registers information and guides the dynamics of
activity patterns.
Borisyuk R & Hoppensteadt F (2004) A theory of epineuronal memory. Neural Networks
Formulas of Epineuronal Memory
Variables x(t); parameters p(t); stochastic process (t).
Mnemonic landscape function M(t,x,p,).
x = f (t , x, t ,  )
 p  p =   p M (t , x, p,  ).
Reaction-diffusion equation for the landscape function M(t,x,p)
M
=  22x , p M (t, x, p,  )  S (t, x, p)  g (t, M )
t
Input (source) term represents a pattern for storage u * :
S (t , x, p ) = u *   ( x  u * )
Example of the reaction term :
g (t , M ) =  M ( M   )( M   )
Borisyuk R & Hoppensteadt F (2004) A theory of epineuronal memory. Neural Networks
Memory of 15 random mnemes
REWARD15
REWARD1
REWARD2
Recall Starting From Random Initial Data
Example of recall
Uniform distribution
between 15 memorised
mnemes. Histogram of
1000 recalls starting
from random initial data
Recall of Five Sequential Patterns
The landscape function peak heights indicate the sequential
order of recall
Epineuronal Memory: 5 Peaks Mnemonic Surface
Mnemonic function M(u)
1D vector x
-11.7
-11.75
x
-11.8
20
dx/dt
-11.85
15
-11.9
-11.95
10
-12
5.3
5
5.305
5.31
5.315
5.32
5.325
5.33
5.335
5.34
5.345
ZOOM
0
-5
Complex dynamics
-10
Dynamical uncertainty
-15
-20
0
1
2
3
4
5
6
x
Mnemonic Landscape and Trajectories
Borisyuk R & Hoppensteadt F (2004) A theory of epineuronal memory. Neural Networks
Conclusions
• Study of chain model of the hippocampus shows that phase shift
between two inputs controls spatio-temporal patterns (hippocampal
bar code)
• Phase shift, synchronization and resonance have been used to
memorise signals and detect their novelty without modification of
synaptic strengths
• STDP-type learning rule has been used to memorise sequences and
replay them in forward and reverse order
• General theory of epineuronal memory has been developed which
includes both phase-shift and STDP based memories.
• The epineuronal paradigm demonstrates mechanisms for stable and
persistent memory in the presence of noisy and uncertain
environments. It introduces the mnemonic landscape that governs
regulation of a brain structures. This approach enables the
memorization of events and sequences of events.
END
PLYMOUTH
Happy Birthday to Frank!