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

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Multiple attractors and transient synchrony in a
model for an insect's antennal lobe
Joint work with B. Smith, W. Just and S. Ahn
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Olfaction
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Schematic of the bee olfactory system
Antennal lobe
Local interneurons (LNs)
Output
Input from receptors
Projection neurons (PNs)
Glomeruli (glom) sites of synaptic contacts
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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.

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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
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PNs respond differently to the same odor
(Laurent, J. Neuro.96)
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Transient Synchronization of Spikes
(Laurent, TINS 96)
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What is the role of transient synchrony?

• Is the entire sequence of dynamic clusters
  important?

• Decorrelation of inputs (Laurent)

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Goal Construct an excitatory-inhibitory
network that exhibits

• Transient synchrony

• Large number of attractors/transients

• Decorrelation of inputs

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The Model
ASSUME PNs can excite one another
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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)
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Discrete Dynamics
This solution exhibits transient synchrony
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Discrete Dynamics
Network Architecture
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What is the complete graph of the dynamics?
How many attractors and transients are there?
Network architecture
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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 ?

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Numerics
-- There is a phase transition at sparse
coupling. -- There are a huge number of stable
attractors if probability of coupling is
sufficiently large
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Length of transients
Length of attractors
? fraction of cells with refractory period 2
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Rigorous analysis

• When can we reduce the differential equations
  model
• to the discrete model?

2) What can we prove about the discrete model?
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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.
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100 Cells - Each cell connected to 9 cells
Cell number
Cell number
time
Discrete model
ODE model
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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

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Question How prevalent are minimal cycles?
Does a randomly chosen state belong to a minimal
cycle?
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Let ?(n) probability of connection.
The following result states that there is a
phase transition when ?(n) ln(n) / n
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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 ? ?.
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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.
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Theorem Assume that each pi lt p and ?i lt ?.
Fix ? ? (0,1). Then ? C(p, ?, ?) such that
if ?(n) gt 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.