Fluctuation-Driven Rhythms in
Neuronal Networks
Alla Borisyuk
University of Utah
Plan
• Biological examples of rhythms
• Model I: Noise as kindling
- building the model(s)
- analysis: model reduction
- benefits of noise
• Model II: Rhythm from a “noisy crowd”
- building the model
- analysis: parameter constraints
- (unexpected) predictions
Example 1: Pre-Bötzinger complex
• Area in brainstem
Pre-Bötzinger complex
• Area in brainstem
• Control of respiratory rhythm
From Del Negro et al. 2001
Pre-Bötzinger complex
• Area in brainstem
• Control of respiratory rhythm
• Frequency can be modulated over large range
Pre-Bötzinger complex
• Area in brainstem
• Control of respiratory rhythm
• Frequency can be modulated over large range
Mechanisms?
Driven by a pacemaker (oscillator)?
• ~5% of neurons are pacemakers
• Blocking putative pacemaking mechanism
preserves the rhythm
Pre-Bötzinger complex
• Area in brainstem
• Control of respiratory rhythm
• Frequency can be modulated over large range
Mechanisms?
Emergent network property?
Role of noise?
Emergent network property
Feldman and Del Negro 2006
Emergent network property
Role of noise as a control mechanism?
Large range of frequencies?
Feldman and Del Negro 2006
Example 2:
Developing spinal cord
• Population oscillations (O’Donovan and Chub 1997)
Developing spinal cord
• Population oscillations (O’Donovan and Chub 1997)
• No bursters, pacemakers <5%
• Episode builds up from individual events
Developing spinal cord
•
•
•
•
•
Population oscillations (O’Donovan and Chub 1997)
No bursters, pacemakers <5%
Episode builds up from individual events
Effectively excitatory connections
Role of synaptic depression or spike-frequency
adaptation
• Explored in ad-hoc firing rate models
(Tabak et al. 2000, 2001) and spiking models with
heterogeneity (Vladimirski et al., in press)
Developing spinal cord
•
•
•
•
•
Population oscillations (O’Donovan and Chub 1997)
No bursters, pacemakers <5%
Episode builds up from individual events
Effectively excitatory connections
Role of synaptic depression or spike-frequency
adaptation
• Explored in ad-hoc firing rate models
(Tabak et al. 2000, 2001) and spiking models with
heterogeneity (Vladimirski et al., in press)
Same model as Pre-Botzinger complex
Example 3: Insect Antennal Lobe
Mushroom body
Lateral protocerebrum
Antennal lobe
Antenna: 60 000 receptors
• Analogue of the olfactory bulb in mammals
• First stage of processing of the olfactory input
• Activity of cells represents olfactory stimuli
(spatially and temporally)
www.neurobiologie.fu-berlin.de/galizia/
Properties of the rhythm
Individual Neurons:
no oscillation in spike trains
Power spectrum
Population: oscillation
20-30 Hz
No peak in power spectrum
25 Hz
Consequently: low correlation between population and ind. neurons
Bazhenov et al. 2001
Properties of the rhythm
Individual Neurons:
no oscillation in spike trains
Power spectrum
Population: oscillation
20-30 Hz
No peak in power spectrum
25 Hz
Subthreshold voltage
is more oscillatory
Bazhenov et al. 2001
Properties of the rhythm
Individual Neurons:
no oscillation in spike trains
Power spectrum
Population: oscillation
20-30 Hz
No peak in power spectrum
25 Hz
Mechanisms?
Subthreshold voltage
is more oscillatory
Bazhenov et al. 2001
The Role of the Antennal Lobe
• First stage of processing of the olfactory input
• Spatial, rate, and temporal coding of odors
• Odor representation in the AL changes with
learning
Odor 1
Before
training
food!
After
training
Odor 2
The Role of the Antennal Lobe
• First stage of processing of the olfactory input
• Spatial (rate), and temporal coding of odors
• Odor representation in
the AL changes with
col
learning
Odor 1
Odor 2
Before
training
food!
After
training
color = activity level
The Role of the Antennal Lobe
• First stage of processing of the olfactory input
• Spatial, rate, and temporal coding of odors
• Odor representation in the AL changes with
learning
Odor 1
Before
training
food!
After
training
Odor 2
The Role of the Antennal Lobe
• First stage of processing of the olfactory input
• Spatial, rate, and temporal coding of odors
• Odor representation in the AL changes with
learning
• AL network itself is modified with learning
(Farooqui et al 2003)
The Role of the Antennal Lobe
• First stage of processing of the olfactory input
• Spatial, rate, and temporal coding of odors
• Odor representation in the AL changes with
learning
• AL network itself is modified with learning
(Farooqui et al 2003)
What changes in the network?
Model I: Noise as kindling
• Noisy spiking model can be used to produce
the rhythm?
• Analyze using mean-field reduction
• Noise amplitude controls the frequency of
oscillations
Spiking models
Leaky integrate-and-fire with hyperpolarization
A. Synaptically-driven
B. Calcium-mediated
• Reduce to a mean-field description
• Look for oscillatory solutions
Nesse, Borisyuk, Bressloff (2008)
Spiking models
Leaky integrate-and-fire with hyperpolarization
A. Synaptically-driven
B. Calcium-mediated
Spiking models
Below threshold
fast
Adaptation
v
v I h s(t ) x(t )
External inputs (noise)
v(t )
v(t
)
v
r
r
v
Network connections
Spiking (reset)
h is slow and different in two models
(introduced below)
Individual cells
No intrinsic bursts
Connections

All-to-all connections: each cell receives the same
population input
s
s

sw
Each spike produces an
alpha-function

s w
as
w
N
(t ti j )
i
j
Noise

External inputs modeled as white noise arriving at
synapses with time constant x
(t)
x
x
x y
x
y
y
x
(t )
A: Synaptically-driven adaptation

h
h
slow
h ah ( s(t ) x(t ))
A: Synaptically-driven adaptation
v i
v
h
vi
I hi
h i
hi
ah ( s xi )
s
s w
s

sw
as
w
N
x i
x
xi
y i
yi
y
s xi ; vi (t )
(t ti j )
i
j
yi
x
i
(t )
vi (t
r
) vr
Population bursting
Spikes build-up
Increase in noise amplitude
increases burst frequency
Further increase in noise amplitude
abolishes bursting
Noise controls bursting
Building mean-field model
v
h
v i
vi
I hi
h i
hi
ah ( s xi )
s
s

sw
<x>=0
τv« τs, τx« τh , large N
input-output (f-I) curve f(x)
s w
as
w
N
x i
x
xi
y i
yi
y
f(I-h+s+x)
s xi
(t ti j )
i
j
yi
x
i
(t )
<f> ensemble average
Building mean-field model

h
h

ss
w
f
w
h ah s
s w
w as f
f ( I h s x) p( x)dx
with p( x) - Gaussian with st. dev. / 2
Ornstein-Uhlenbeck process
x
x
x
x
y
y
y
x
(t )
p(x)
A: Mean-field model
• 3-dimensional model

h
h

ss
w
w
h ah s
s w
w as f
• Find steady states (between 1 and 3)
• Linear stability
Bifurcation diagram (schematic)
s
σ
Varying noise and I
Varying noise
Varying I
Burst rate
2.5
0.5
0.75
B: Calcium-mediated adaptation
slow

h
h
h h (c )
c
c ac
c
(t t j )
j
B: Mean-field model
• 4-dimensional model

h
h

cc
s

ww
s
h h (c )
c
f (s h I )
s w
w as f
Produces bursting
Onset of bursting through SNIC
• Even larger range of burst frequencies
Conclusions
• Population bursting exists for a range of noise
amplitudes
• Mean-field behavior matches the spiking
model
• Onset of periodic solution through Hopf
bifurcation or SNIC, depending on details of
spiking model
• Frequency can be controlled by changing σ
• The model can be used for the spinal cord too
Model II: Rhythm from a “noisy crowd”
Antennal Lobe
•
•
•
•
E,I cells (800, 4000 bee; 900, 300 locust)
Connectivity is not well known (E→E ?)
Some I cells have wide arborization
Fast excitation (Ach), fast inhibition (GABAA), slow
inhibition (hystamine?)
Oscillations and odor coding
• Mechanisms proposed in earlier models (e.g.
Bazhenov et al. 2003)
• Also included: sequence of E ensembles as a
stimulus code
Oscillations and odor coding
• Mechanisms proposed in earlier models (e.g.
Bazhenov et al. 2003)
• Also included: sequence of E ensembles as a
stimulus code
• Alternatively: mostly spatial code, oscillations
as bi-product of the network activity
Model
Theory: (Brunel,Hakim(1999); Brunel,Wang(2003))
• Leaky integrate-and-fire (LIF)
• Random connections with fixed probability
• Synaptic delays
• Poisson excitatory external input
• Assume asynchronous state (constant population
activity), look for onset of oscillations
• Near onset: population oscillation with noisy
neurons
Numerics: LIF, Morris-Lecar
Tuning the model
Looking for onset of oscillations, i.e. assume
νE,I(t)= ν0E,I(1+εE,Ieiωt)
Brunel and Wang, 2003
Tuning the model
Looking for onset of oscillations, i.e. assume
νE,I(t)= ν0E,I(1+εE,Ieiωt)
νE
νI
Brunel and Wang, 2003
Tuning the model
Looking for onset of oscillations, i.e. assume
νE,I(t)= ν0E,I(1+εE,Ieiωt)
ν E → sE
νI
→ sI
Brunel and Wang, 2003
Tuning the model
Looking for onset of oscillations, i.e. assume
νE,I(t)= ν0E,I(1+εE,Ieiωt)
νE → sE → Isyn,E
νI
→ sI → Isyn,I
Brunel and Wang, 2003
Tuning the model
Looking for onset of oscillations, i.e. assume
νE,I(t)= ν0E,I(1+εE,Ieiωt)
νE → sE → Isyn,E
→ νE
νI
→ νI
→ sI → Isyn,I
Iext
Brunel and Wang, 2003
E
I
νE,I(t)= ν0E,I(1+εE,Ieiωt)
Equation for frequency:
ΦI
Λ sin ΦE
atan
Λ cos ΦE
1
E
I
νE,I(t)= ν0E,I(1+εE,Ieiωt)
Equation for frequency:
ΦI
Λ sin ΦE
atan
Λ cos ΦE
1
where Φ ω depend on synaptic time constants
Φω
ωτl
atan ωτ r
atan ωτ d
E
I
νE,I(t)= ν0E,I(1+εE,Ieiωt)
Equation for frequency:
ΦI
Λ sin ΦE
atan
Λ cos ΦE
1
where Φ ω depend on synaptic time constants
Φω
ωτl
atan ωτ r
atan ωτ d
E
I
νE,I(t)= ν0E,I(1+εE,Ieiωt)
Equation for frequency:
ΦI
Λ sin ΦE
atan
Λ cos ΦE
1
where Φ ω depend on synaptic time constants,
Λ depends on the relative strength of connections
Λ
X IE X EI /X II
E
Λ sin ΦE
atan
Λ cos ΦE
ΦI
I
1
Biological constraints:
Phase: population I lags by π/2
Frequency: f = ω/(2π) = 20 to 30 Hz
ΦI 2
25
ωτlI
atan ωτ rI
using atan(x)<x
atan ωτ dI
2
E
I
Λ sin ΦE
atan
Λ cos ΦE
ΦI
1
Biological constraints:
Phase: population I lags by π/2
Frequency: f = ω/(2π) = 20 to 30 Hz
ΦI 2
25
ωτlI
τ lI
atan ωτ rI
τ rI
τ dI
atan ωτ dI
10 msec
2
Need: τ lI τ rI τ dI 10 msec
Biological estimate: τ lI τ rI τ dI
Firing rate is too high
E
I
7 msec
Need: τ lI τ rI τ dI 10 msec
Biological estimate: τ lI τ rI τ dI
Firing rate is too high
E
7 msec
I
Solution: Add E-E connections
Solve equations for frequency numerically
E-E connections reduce oscillations rate
RHS
Prediction: E-E connections
without E-E
with E-E
frequency
Blocking inhibition
• Reducing strength of inhibition abolishes the
oscillations (asynchronous firing is the only
solution)
• Biologically: MacLeod & Laurent (1996),
Mwilaria et al (2008)
Change in odor concentration
Amplitude
Frequency
• Biologically: Increasing concentration
increases amplitude of oscillations, but leaves
frequency unchanged (Stopfer et al. 2003)
• Concentration ≈ Firing rate of input
input
input
Short-term memory
With repeated presentations of a stimulus
• Cells are more synchronized
• Firing rate of individual cells decreases
• Oscillation frequency remains the same
(Stopfer, Laurent 1999)
What changes in network connectivity
could be causing these?
Short-term memory
ΦI
Λ sin ΦE
atan
Λ cos ΦE
Frequency depends on Λ
1
X IE X EI /X II
X EE
X IE
In E-E network:
and
X EI
X II
On the other hand, strength of connection is a
bifurcation parameter
Prediction: learning increases strength of both
excitation and inhibition, keeping the balance
Slow patterns
Biologically: Stimulus- and neuron- specific
slow patterns of activity and inactivity;
picrotoxin-incensitive
Bazhenov et al. 2001
Slow patterns
Biologically: Stimulus- and neuron- specific
slow patterns of activity and inactivity;
picrotoxin-incensitive
Suggestion: Neurons are driven between “ON”
and “OFF” states (perhaps bistable?) by
fluctuations in the receptor neurons’ activity
A neuron is able to participate in the network
activity only during the “ON” state
Summary
Properties:
• Oscillatory population, non-oscillatory cells
• Frequency 20-30 Hz, phase lag between populations π/2
• Oscillations fast-inhibition-dependent
• Population frequency is independent of concentration,
amplitude is increasing
Predictions:
• Subthreshold voltage oscillations in synchrony with
population oscillation
• E-E connections
• Learning increases strength of both excitation and inhibition
keeping the balance
• Bistable neurons with receptor neurons’ activity acting as
switch
Finally,
Noise can be beneficial for the rhythms