Oscillatory Models of Hippocampal Activity and

MemoryRoman 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

Publications1. 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 ActivityWe 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.

V(t+1 ) V(t+N )

V(t+1 ) V(t+N )

Septum

Entorhinal Cortex

S1 SN . . .

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 inputsC –S controls patterns of activity

Gamma and Theta Rhythms of Single Oscillator

0 200 400 t

Single oscillator under influencesof two inputs can demonstrate complex behaviour with slow (theta) and fast (gamma) components

Recoding from hippocampal population(Van Quyen & Bragin, 2007)

Spatio-Temporal Patterns Hippocampal Bar Code

Borisyuk & Hoppensteadt, 1999, Biological Cybernetics

HIPPOCAMPUS

Phase deviation is a key parameter which coltrols dynamics of hippocampal

activity

TIME TIME

SEPTAL)2sin( ft

Input:EC

)2sin( ft

Input:

=5 =18

Phase/Frequency Coding and Novelty Detection

• Equations of ONN dynamics:

q

l

jk

jl

jl

n

i

jkij

jk

jk ag

q

wt

n

v

dt

d

11

10 )sin()()2sin(2

n

i

jkij

jk

jk t

nga

dt

da

10

22 )(cos

1

dt

dag

dt

d jkj

kj

k

jk

)(1

)2,1(,)/)exp((1

)/)exp(()(

ix

xxg

ii

iii

Borisyuk, Denham, Kazanovich, Hoppensteadt, Vinogradova

(2000,2001)

Model Description

Dynamics of oscillator’ frequencies is governed by the learning rule: here we do not modify connection strengths, instead we adjust natural frequency

0 500 1000 1500 2000 2500

time4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9Naturalfrequencies

Stimulation

G1

G2

G3

=5 =7 =8

Dynamics of Frequencies and Amplitudes

Am

pli

tud

e

Time

#

6,4

6,6

6,8

7

7,2

7,4

7,6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time

Na

tura

l fre

qu

en

cie

s

6,4

6,6

6,8

7

7,2

7,4

7,6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time

Nat

ura

l fre

qu

en

cie

s

Resonant state Non-Resonant state

Novelty Detection: Sparse Coding

The bar’s height is proportional to the number of resonant oscillators in the group.

The arrow indicates coincidence of resonant oscillator groups for the same symbols “O”

O

H

E

L2

L1

W

O

D

R

L

0 2000

Example of sparse coding: 10 objectare coded by 2000 groups

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)

j nw n,j

Tm+1Tm

Activity

))1(())(()(

)1()2(

maxmax

,,

hmEhmE

mwmwnj

jnjn

Time

Threshold h

Borisyuk Denham, Denham, Hoppensteadt, 1999, Rev in Neurosc.

Oscillatory Memory

 

0.2 0.4 0.6time

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.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

t  

 

 

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

t  

ONN Memory: Sequence of 5 Patterns

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)

Foster & Wilson, Nature, 2006

Place cell 1 fires

Place cell 2 fires

Place cell 3 fires

Reverse replay

Reverse Replay with Anti STDP

Reverse Replay with Anti STDP

Forward and Reverse ReplayA 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 shownin 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

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.

Forward preplay Reverse replay

Diba & Buzsaki, Nature Neurosc 2007

Epineuronal MemoryA 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

).,,,(

),,,(

pxtMpp

txtfx

p

))((),(

:termreaction theof Example

)(),,(

:storageforpatterna representsterm(source)Input**

*

MMMMtg

uxupxtS

u

Variables x(t); parameters p(t); stochastic process (t).Mnemonic landscape function M(t,x,p,).

),(),,(),,,(2

,

2 MtgpxtSpxtMt

Mpx

Reaction-diffusion equation for the landscape function M(t,x,p)

Borisyuk R & Hoppensteadt F (2004) A theory of epineuronal memory. Neural Networks

Memory of 15 random mnemes

REWARD2

REWARD1

REWARD15

Recall Starting From Random Initial Data

Uniform distribution between 15 memorised mnemes. Histogram of 1000 recalls starting from random initial data

Example of recall

Recall of Five Sequential Patterns

The landscape function peak heights indicate the sequential order of recall

Epineuronal Memory: 5 Peaks Mnemonic Surface

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

5.3 5.305 5.31 5.315 5.32 5.325 5.33 5.335 5.34 5.345-12

-11.95

-11.9

-11.85

-11.8

-11.75

-11.7

ZOOM

1D vector x

x

dx/dt

Complex dynamics

Dynamical uncertainty

Mnemonic function M(u)

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!Happy Birthday to Frank!