John Wordsworth, Peter Ashwin,Gabor Orosz, Stuart Townley

Mathematics Research InstituteUniversity of Exeter

Overview1. Motivation & Base Model

1. Olfaction, the Honey Bee, Objectives.2. The Model, Core Dynamics and Spatial Coding.

2. Stimuli and Spatio-Temporal Codes1. Adding Input and Temporal Coding.2. Temporal Coding and Spatio-Temporal Codes.

3. Future Work and Uses1. Other Cases and Introducing Noise.2. Classifying Inputs from a Spatio-Temporal Code.

1. Motivation and Base Model

A brief look at background and the system of coupled phase oscillators at the heart of

this work.

MotivationHoney Bee experiments show neurons in the Glomeruli fire in clusters;

C. Galizia, S. Sachse, A. Rappert & R. Menzel, 1999

ObjectivesSpatio-Temporal Coding Emulate Olfaction

• Classify olfactory information in an interesting fashion.

• Suggest components of combined olfactory encoding.

Steady Stimuli Input

Coupled Oscillator Model

Distinct Coding Output

Phase (Ѳ), Natural Frequencies (ωi), Coupling Strength (K), Number of Oscillators (N), Coupling Function (g), Noise (η), Stimuli (X).

Global All-to-AllCoupling

Phase Oscillator Model

Dynamics (with Noise)Synchrony

(Alter Alpha)

Anti-synchrony

Clustering

Chaos

Synchrony(Alter Alpha)

Anti-synchrony

Clustering

Chaos

Dynamics (with Noise)

Synchrony

Anti-synchrony(Alter Beta)

Clustering

Chaos

Dynamics (with Noise)

Synchrony

Anti-synchrony

Clustering(Alpha:1.7, Beta: -2)

Chaos

Dynamics (with Noise)

After Transients, we have; Yellow ‘Stable’ Cluster. Blue ‘Unstable’ Cluster. One Lone Oscillator.

Cluster States (with Noise)

What Happens?1. Initial Transient.2. First Switches Fast.3. Residence Time Increases.4. System Stalls.

Note;Considered as a Neural System -the system is still firing!

Memory Effect;System does not change unless stimulated.

Cluster States (Without Noise)

Residence times increase exponentially.The system becomes stalled.

We need some form of stimuli to force the system.

Cluster States (Without Noise)

Time

Oscillator 5

Oscillator 4

Oscillator 3

Oscillator 2

Oscillator 1

Linearizing around a state and taking Eigenvalues;

Yellow is ‘stable’;Yellow -> Blue

White -> Yellow

Blue is ‘unstable’;Blue -> ?

Turns out that the‘fastest’ blue oscillator -> white.

Spatial Coding

Heteroclinic Network; 30 Cluster States, 60 Orbits.Or a Directed Graph?

Sample Coding; BBWYY (S1) WYYBB (S2) YBBYW (S3) BWYBW (S4)

A Network of Cluster States

Summary of Spatial Codes• System of 5 coupled phase oscillators.• Find (2,2,1) ‘clusters’ for certain parameters.• Only 2 possible states at the next step once at a

given state – which one is random when the system is driven by noise.

• This generates a system with 30 states and 60 connections.

• A spatial code can be seen as a series of state identifiers.

2. Stimuli & Spatio-Temporal Coding

Next we add input to the System and generate a temporal code which we can

combine with our spatial coding.

Using different frequencies as input.

Detuning Frequencies as Input

Where Delta is the Amplitude of Detuning (Strength of Odor?)

Uniform Detuning (No Noise)

Regular Residence Times, Repeated Pattern.

Time

Oscillator 5

Oscillator4

Oscillator 3

Oscillator 2

Oscillator 1

Spatial Coding of the Detuned System

Reduced State Graph / Spatio-Coding of Data

Spatial Coding of the Detuned System

Large Amplitude of Detuning (Top), Very Small Amplitude (Bottom)

Temporal Coding of the Detuned System

Just to prove we’re still thinking about Spiking

States are a statement of when spikes are together.Rethink the meaning of residence times

A Neuroscience View

Post transient residence times are fixed (Red).

Evaluation of Residence Times

Four ‘levels’ of residence times, not 6 – lost info.

Evaluation of Residence Times

Direct relation between residence times and δ.Can determine δ from code.

Detuning Amplitude vs. Residence Times

Original Graph driven by noise;

Combined Spatio-Temporal Coding

Spatio-Temporal Coding of Detuned Inputs

Large Amplitude of Detuning (Top), Very Small Amplitude (Bottom)

Spatio-Temporal Coding

Summary of Spatio-Temporal Coding• Detune the frequencies of the oscillators as a

method of inputting data to the system.• Alters the spatial coding by removing edges that the

system can follow (minimal amount remain with uniform detuning).

• Differences in the magnitudes of frequencies determines residence times (temporal code).

• Should be able to determine input frequencies from a given spatio-temporal code.

3. Future Work and Application

Finally, a brief look at work that we are currently focussing on and some

applications of the work.

Alternative Detunings

If we run the system with both Detuning and Bounded Noise, which wins? Compare residence time-windows.

Residence Time

Case 3

Case 2

Case 1 Detuning Noise

Noise Detuning

NoiseDetuning

Detuning & Noise Together

Interim Cases• Non-Periodic cycles appear when we have noise and

certain types of detuning. • Have seen cases with 2 potential cycles of 6, in which

the system traverses one a seemingly random number of times then the other once.

• Essentially a sliding scale for number of paths with probability of traversal > 0 between (a) when uniform detuning is far greater than noise and (b) when there is no detuning and just noise.

• These are the most interesting cases

Classifying Input from a given Code• Consider Spatial and Temporal components.

• Temporal Component will dictate magnitude of differences between natural frequencies.

• Spatial Component will dictate ordering of natural frequencies.

• Could be complex for non-uniform detunings.

• We can generate one of two unique spatio-temporal code / firing pattern for a given input, depending on initial conditions.

• Non-Uniform detunings -> Other Patterns.• Analysis of residence times can tell us whether noise

or detuning is driving the system.• Data is lost when considering spatial or temporal

code alone, but is probably complete using both combined.

Conclusion

Peter Ashwin,Stuart Townley,Gabor Orosz,University of Exeter

Thank You