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

1. Multiple attractors and transient synchrony in a model for an insect's antennal lobe Joint work with B. Smith, W. Just and S. Ahn

2. Olfaction

3. Schematic of the bee olfactory system Antennal lobe Local interneurons (LNs) Output Input from receptors Projection neurons (PNs) Glomeruli (glom): sites of synaptic contacts

4. Neural Coding in OB/AL • Each olfactory sensory cell expresses one of • ~200 receptors (~50000 sensory cells) • 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.

5. Data: spatial and temporal Imaging Single cell/population • Population activity exhibits • approx. 30 Hz oscillations • Different odors activate • different areas of antennal lobe • Individual cells exhibit • transient synchronization • (dynamic clustering) Pentanol Orange oil • Odorants with similar molecular structures activate overlapping areas www.neurobiologie.fu-berlin.de/galizia/ Stopfer et al., Nature 1997

6. PN’s respond differently to the same odor (Laurent, J. Neuro.‘96)

7. Transient Synchronization of Spikes (Laurent, TINS ‘96)

8. What is the role of transient synchrony? • Is the entire sequence of dynamic clusters important? • “Decorrelation” of inputs(Laurent)

9. Transient phase may be more important than attractor. 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. (Mazor, Laurant, Neuon 2005)

10. Goal: Construct an excitatory-inhibitory network that exhibits: • Transient synchrony • Large number of attractors/transients • Decorrelation of inputs

11. - directly - via interneurons - via rebound The Model ASSUME: PN’s can excite one another

12. 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) Assume: A cell does not fire in consecutive episodes (1,4,5)

13. (1,6) (4,5) (2,3,7) (1,5,6) 1 fires with 5 and 6 (2,4,7) (3,6) 1 fires with 4 and 6 (1,4,5) Discrete Dynamics This solution exhibits transient synchrony

14. (1,2,5) (1,6) (1,3,7) (4,6,7) (4,5) (4,5,6) (2,3,5) (2,3,7) (1,6,7) (1,5,6) (3,4,5) (2,4,7) (1,2,7) (3,6) (3,4,5,6) (1,4,5) Different transcient Same attractor Different transcient Different attractor Discrete Dynamics Network Architecture

15. 2 7 6 5 4 3 1 What is the complete graph of the dynamics? How many attractors and transients are there? Network architecture

17. 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 ?

18. Numerics 2000 Number of attractors Number of connections per cell 5 10 -- There is a “phase transition” at sparse coupling. -- There are a huge number of stable attractors if probability of coupling is sufficiently large

19.  = .5  = .5  = 0  = 0 Length of transients Length of attractors  = fraction of cells with refractory period 2

20. Rigorous analysis When can we reduce the differential equations model to the discrete model? 2) What can we prove about the discrete model?

21. 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: No - for purely inhibitory networks. Yes - for excitatory-inhibitory networks.

22. 100 Cells - Each cell connected to 9 cells Cell number Cell number time Discrete model ODE model

23. Rigorous analysis of Discrete Dynamics We have so far assumed that: Refractory period = p If a cell fires then it must wait p episode before it 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

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

25. Example: 2 7 Indegree of vertex 5 = 3 Need some notation: Outdegree of vertex 5 = 2 6 5 4 3 1 Let (n) = probability of connection. The following result states that there is a “phase transition” when (n) ~ ln(n) / n

26. A phase transition occurs when (n) ~ ln n / n. The following result suggests  another phase transition ~ C/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 .

27. 1357 MC = {5,7} 246 1246 357 126 3457 347 12356 26 457 1236 1256 12346 57 236 47 256 1347 12356 1457 MC = {4,7} 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}.

28. 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.