Multiple attractors and transient synchrony in a model for an insect's antennal lobe

Joint work with B. Smith, W. Just and S. Ahn

Olfaction

Schematic of the bee olfactory system

Antennal lobe

Input from receptors

Output

Local interneurons (LNs)

Projection neurons (PNs)Glomeruli (glom):sites of synaptic contacts

• Each olfactory sensory cell expresses one of ~200 receptors (~50000 sensory cells)

Neural Coding in OB/AL

• Sensory cells that express the same receptor project to the same glomerulus

• Each odorant is represented by a unique combination of activated modules.

• Highly predictive relationship between molecules, neural responses and perception.

Data: spatial and temporal

Orange oil Pentanol

Imaging Single cell/population

• Odorants with similar molecular structures activate overlapping areas

www.neurobiologie.fu-berlin.de/galizia/ Stopfer et al., Nature 1997

• Population activity exhibits approx. 30 Hz oscillations

• Individual cells exhibit transient synchronization(dynamic clustering)

• Different odors activatedifferent areas of antennal lobe

PN’s respond differently to the same odor

(Laurent, J. Neuro.‘96)

Transient Synchronization of Spikes

(Laurent, TINS ‘96)

What is the role of transient synchrony?

• Is the entire sequence of dynamic clusters important?

• “Decorrelation” of inputs (Laurent)

Neural activity patterns that represent odorants in the AL are statistically most separable at some point during the transient phase, well before they reach a final stable attractor.

Transient phase may be more important than attractor.

(Mazor, Laurant, Neuon 2005)

Goal: Construct an excitatory-inhibitory network that exhibits:

• Transient synchrony• Large number of attractors/transients• Decorrelation of inputs

The Model

ASSUME: PN’s can excite one another

- directly

- via interneurons

- via rebound

Transient: linear sequence of activation

Period: stable, cyclic sequence of activation

Reduction to discrete dynamics

(1,6)

(4,5)

(2,3,7)

(1,5,6)

(2,4,7)

(3,6)

(1,4,5)Assume: A cell doesnot fire in consecutiveepisodes

(1,6)

(4,5)

(2,3,7)

(1,5,6)

(2,4,7)

(3,6)

(1,4,5)

This solution exhibits transient synchrony

1 fires with 5 and 6

1 fires with 4 and 6

Discrete Dynamics

Discrete Dynamics

(1,6)

(4,5)

(2,3,7)

(1,5,6)

(2,4,7)

(3,6)

(1,4,5)Different transcient

Same attractor

(1,3,7)

(4,5,6)

Different transcientDifferent attractor

NetworkArchitecture

(1,2,5)

(4,6,7)

(2,3,5)

(1,6,7)

(3,4,5)

(1,2,7)

(3,4,5,6)

What is the complete graph of the dynamics?

How many attractors and transients are there?

2 7

6 5

4

1

3

Network architecture

Analysis

How do the- number of attractors- length of attractors- length of transients

depend on network parameters including - network architecture - refractory period - threshold for firing ?

Numerics

2000

Number ofattractors

Number of connections per cell5 10

-- There is a “phase transition” at sparse coupling.-- There are a huge number of stable attractors if probability of coupling is sufficiently large

= fraction of cells with refractory period 2

Length of transients Length of attractors

= .5 = .5

= 0

= 0

Rigorous analysis

1) When can we reduce the differential equations model to the discrete model?

2) What can we prove about the discrete model?

Reducing the neuronal model to discrete dynamics

Given integers n (size of network) and p (refractory period), can we choose intrinsic and synaptic parameters so that for any network architecture, every orbit of the discrete model can be realized by a stable solution of the neuronal model?

Answer:

- for purely inhibitory networks.No

Yes - for excitatory-inhibitory networks.

100 Cells - Each cell connected to 9 cells

Discrete model ODE model

Cell number Cell number

time

We have so far assumed that:

If a cell fires then it must wait p episode beforeit can fire again.

Threshold = 1

If a cell is ready to fire, then it will fire if it received input from at least one other active cell.

We now assume that:• refractory period of every cell = pi

• threshold for every cell = i

Refractory period = p

Rigorous analysis of Discrete Dynamics

Question: How prevalent are minimal cycles? Does a randomly chosen state belong to a minimal cycle?

1

2 7

6 5

4 3

Need some notation:

Example:

Indegree of vertex 5 = 3

Outdegree of vertex 5 = 2

Let (n) = probability of connection.

The following result states that there is a “phase transition” when (n) ~ ln(n) / n

Theorem 1: Let k(n) be any function such that k(n) - ln(n) / ln(2) as n .Let Dn be any graph such that the indegree of every vertex is greater than k(n). Then the probability that a randomly chosen state lies in a minimal attractor 1 as n .

Theorem 2: Let k(n) be any function such that ln(n) / ln(2) - k(n) as n .Let Dn be any graph such that both the indegree and the outdegree of every vertex is less than k(n). Then the probability that a randomly chosen state lies in a minimal attractor 0 as n .

A phase transition occurs when (n) ~ ln n / n.

The following result suggests another phase transition ~ C/n.

Definition: Let s = [s1, …., sn] be a state. Then MC(s) VD are those neurons i such that si(t) is minimally cycling. That is, si(0), si(1), …, si(t) cycles through {0, …., pi}.

1457

236

457

126

357

246

1357

1246

3457

1236

1256

1347

12356

26

5712346 47 256

34712356

MC = {5,7}

MC = {4,7}

Theorem: Assume that each pi < p and i < . Fix (0,1). Then C(p, , ) such that if (n) > C/n, then with probability tending to one as n , a randomly chosen state s will have MC(s) of size at least n .

That is: Most states have a large set of minimally cycling nodes.