introduction to computational neuroscience: minimal...
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Introduction to Computational Neuroscience:
Minimal Modeling
Amitabha BoseNew Jersey Institute of Technology &
Jawaharlal Nehru University
Collaborators: Farzan Nadim, Yair Manor, Victoria Booth, Jon Rubin, Victor Matveev,
Lakshmi Chandrasekaran, Tim Lewis, Richard Wilson, Yu Zhang
IIMSc – January 2010
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General Biological Questions
• What accounts for the rhythmic activity?• Under what circumstances do synaptic
and intrinsic properties of neurons cooperate or compete?
• What effect do multiple time scales have?• What are the underlying neural
mechanisms that govern behavior?
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Translation to mathematics
• What accounts for the rhythmic activity?Periodic solutions in phase space
• Under what circumstances do synaptic and intrinsic properties of neurons cooperate or compete?
Effect of parameters on solutions• What effect do multiple time scales have?
Singular perturbation theory• What are the underlying neural mechanisms that govern
behavior?Deriving mathematically minimal models that revealnecessary and sufficient conditions
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Short-term synaptic plasticity has been identified in the majority of synapses in the central nervous
system.
From Dittman et al., J. Neurosci, 2000.
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κβ
κβ
reduction
VI
gsyn,i
VI
gsyn,i
α
VI
gsyn,i
Dynamics of short-term synaptic depression
Presynaptic neuron
Presynaptic neuron
===
α
β
κ
τττ Inhibitory decay time constant
Depression time constant
Recovery from depression time constant
Postsynaptic conductance
Postsynaptic conductance
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Plasticity: What’s it good for?In cortical circuits:
• Automatic gain control• Network stabilization• Population rhythm
generation• Direction selectivity• Novelty detection• Coincidence detection• Generation of frequency-
selective responses
In rhythm generating networks:• Multistability• Phase maintenance• Episodic Bursting
Also:• Short-term habituation of the mammalian startle response• Differential stimulus filtering• On-off switching
Abbott, Nelson, Reyes, Markram, TsodyksRinzel, Marder, Nadim etc…
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What advantages does synaptic plasticity confer upon a rhythmic network?
• The answer is architecture dependent.• Plasticity allows different parameters to be relevant to the
network at different frequencies or at different moments in the rhythmic cycle.
• Plasticity can work synergistically with intrinsic neuronal properties.
We will discuss several biologically motivated minimal MATHEMATICAL examples:
We will also discuss in detail one model with no plasticity that leads to an interesting one-dimensional map.
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2-D model for single neuronsBased on Hodgkin-Huxley Formalism
)(/))(('),('
VwVwwwVFV
wτε
−==
∞
-60 -50 -40 -30 -20 -10 0 10 20 30 40 50
0.0
0.1
0.2
0.3
0.4
0.5
w
V (mV)
dV/dt=0 dw/dt=0
applicable becomeson theoryPerturbatiSingular
Geometric 0 As →ε
)(/))(('),(0
Equations Slow
VwVwwwVF
wτ−==
∞ 0/),(/
)( EquationsFast
==
=
ζζ
εζ
ddwwVFddVt
Silent State
Active State
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Model Effect of inhibition
max
( , )
( ) ;( )
( )( )syn
w
pre inh
dV
I g s V
F V wdt
w V wdwdt V
V E
ε
τ∞
−
=
⎧ =⎪⎪⎨ −⎪ =⎪
−⎩
synI
V
• Inhibition lowers the V-nullcline
• If inhibition is strong enough, a curve of “fixed points” is introduced on the inhibited branches.
•Local minima form a jump curve
w
0dwdt
=
0dVdt
=
Curve of fixed pointsVθ
synapticthreshold
Jump curve
Return curve
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Inhibitory SynapseSynaptic current: Isyn= - gmax s (vpost – Esyn), Esyn < vrest
)()(1' prepre VVHsVVHss −−−−= ∞∞ θκ
θγ ττ
Slow Equations in the silent phase
sggss
VwVwwEvsgwVF
inh
w
syn
max
max
/')(/))(('
)(),(0
=−=
−=−−=
∞
κττ
Strong inhibition
Weak inhibition
return curve
S is reset to 1 with a pre-synaptic spike
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Models for depressing synapsesLike other phenomenological reset models of Abbot et al, Markram &Tsodyks
)()(1' θβ
θα ττ
VVHdVVHdd prepre −−−−= ∞∞ Depression
s is reset to the value of d at the moment of a presynaptic spike
dpeak
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Effect of depression on post-synaptic cell
βα
α
ττ
τ
//
/
maxpeak1
1ActInact
Inact
TT
T
eeegg −−
−
−−=
P (but increasing only TInact)
ggeak
gmax
return curve
S is reset to current value of dgmax with each pre-synaptic spike
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Bistability in feedback networks (B., Manor, Nadim, 2001)
Cell controlled Synapse controlled
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Slow manifold for the silent
state of the E neuron
Jump curve
Return curveWe use a one-dimensional map to measures the value of d whenever E is on the return curve
E-cell
depression
I-cell
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Two modes of oscillations• A cell-controlled high frequency mode (weak synapse)
• A synapse-controlled low frequency mode (strong synapse)
Feedback + plasticity forces the synaptic strength to specific values
We can use this observation in other contexts
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Bistablity of anti-phase solutionsMotivated by experimental work of Nadim et al (2001,2005)
Oscillator
1 2
Oscillator
InhibitoryDepressing
Commonly found solutions that arise in such networks are anti-phase spikes.
But what if we could get neuron 1 to fire two spikes in a row?
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Bistability between 1-1 antiphase and 2-2 antiphase firing (Booth, B. 2009)
• Cell 2 is transiently inhibited allowing its synapse to recover more fully.• Stronger inhibition to Cell 1 allows Cell 2 to spike twice.• Cell 1 synapse also recovers
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Bifurcation diagram of n-n antiphasesolutions
1-1
2-2
3-3
4-4
One cell suppressed
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Multistability and clustering in globally inhibitory networks (Chandrasekaran, Matveev, B. 2009)
• Synapses from I to P are inhibitory and depressing
• Synapses from P to I are excitatory
• I fires whenever any Pk fires
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• 1- and 3-cluster solutions also exist
• interspike interval of I decreases with number of clusters
• cluster solutions are periodic
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Periodicity condition in w
is the time it takes the leading cell to reach the jump curve
n = number of clusters (n=2 shown above)
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Periodicity condition in g• If g(0)=g0, what interspike interval tin is
needed to allow g(tin)=g0./actTr e βτ−=
We can derive an n-dimensional map and use linearization to prove stability of solutions
Discrete g values
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Localized Activity in Excitatory Networks(Rubin, B. 2004, also see Drover and Ermentrout)
Bump – no depression
Toggle-switch –with depression
“Weak” stable synapses
Strong synapses
Depressed synapses
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Episodic bursting in respiratory systemsMotivated by experimental work of Wilson (B., Lewis, Wilson, 2005)
• Spontaneous population bursts are observed in cortex, hippocampus etc. and also during development (chick spinal cord, tadpole breathing)
• Tsodyks et al and O’Donovan & Rinzel’sgroup have models for such behavior which involve synaptic plasticity.
• Our example is really, really basic.
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Lung EpisodeBuccal Episode
Overlaid traces show that the lung bursts beginat same point in buccal cycleas in PIR.
Overlaid traces show that the buccal cycles continue predictably from last lung burst.
Experimental ObservationsWilson et al 2002
Lung Area
Buccal Area
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A biologically constrained minimal model
B L
Facilitatinginhibitory synapse
Excitatory synapse
vL = lung voltage, vB = buccal voltagesL = excitatory synaptic gatingsB = inhibitory synaptic gating, dB = facilitation/de-facilitation gating
Morris-Lecar type equations
w
v
vL′ = - Iion - ginhsB[vL-Einh]wL′= [w∞(vL)– wL]/τ (vL)vB′ = - Iion - gexcsL[vB-Eexc]wB′= [w∞(vB)– wB]/τ (vB)sL′ = ν[1-sL]s∞(vL) - ηsL[1-s∞(vL)]sB′ = γ[dB-sB]s∞(vB) - κsB[1-s∞(vB)]dB′ = [1-dB]d∞(vB)/τα - dB[1-d∞(vB)]/τβ
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Bistability of Lung Oscillator
• System undergoes a subcritical Hopf-Bifurcation• Oscillations arise/disappear via saddle-node bifurcations• In between bifurcation points, there is a region of bistability• Nothing special about Iapp
Iapp
Fixed Points
Periodic branch
HB SN
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Episodic nature of lung and buccal driven ventilation
VL= blackVB = gray
dB= blacksB = gray
TL determined by the facilitation time constant τα. TB determined by de-facilitation time constant τβ.
B L
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Remarks about feedback networks
• Plasticity allows regeneration of synaptic currents.
• Plasticity plays a role in setting the network frequency and thereby forces the synapse to operate at a strength dictated by that frequency.
• Plasticity allows certain parameters to be relevant for certain solutions (synapse controlled or cell controlled)
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Feedforward Networks
• Often these are driven by a pacemaker or external inputs.
• By controlling the period of these inputs, the level of plasticity in the network can be fine tuned – differs from the feedback case.
Period of pacemaker
gmax
Synaptic strength
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Phase control of the pyloric rhythm(Cancer borealis)
1t∆
P
2t∆1t∆
Pt
Pt
22
11
∆=
∆=
φ
φ Phase of LP and PY remains constant as the period ofthe pacemaker PD changes (Hooper, 94,95).
What role does plasticity play in setting phase?
PD
LP PY
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O F
Oscillator Follower
Inhibition
Pt∆=φ
Which parameters determine the time ∆ t at which F fires?
Simple premise: ∆ t should be a function of synaptic strength.
(Manor,B., Booth, Nadim, 2004)
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Depressing vs. non-depressing synapse in an O-F network
P = 600 msec P = 1200 msec
• Period P changed by varying O’s interburst interval
O
F no-dep
F dep
Postsynaptic conductanceno-dep and dep
dep
no-dep
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Phase vs. period in O-F networkFor non-depressing synapse:• Time delay ∆t is constant• Phase φ = ∆t / P decreases like
1/P
For depressing synapse:• At small P (weak inhibition),
intrinsic properties of F determine φ
• At larger P (stronger inhibition), φ determined by synaptic properties
φ
*1
1
/2
/peak1
//
/
maxpeak
gecegcee
egg
w
ActInact
Inact
tt
TT
T
=+−
−=
∆−∆−
−−
−
ττ
ττ
τ
κ
βα
α
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Remarks about feedforward networks
• In a feed forward network, the period of the oscillator and thus the strength of its synapse can be fine tuned.
• For phase control, without depression, phase can be chosen at exactly one value of the period. Depression creates a range of periods for which the phase of the postsynaptic cell can be assigned.
• Plasticity again allows certain parameters, intrinsic or synaptic, to be relevant to the network at different frequencies.
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Mathematically intersting example based on pyloric CPG
• Oscillator-follower network governed by inhibition without plasticity
• PY contains an A-current that delays the first spike of the burst
• What effect does A-current have on the dynamics of the feed-forward network?
• Joint work with Farzan Nadim and Yu Zhang 2009
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Morris-Lecar Model for F( , )
( ) ( )
( , ) [ ] ( )[ ] [ ]
w
ext L L
Ca Ca K K
dv f v wdt
w v wdwdt vf v w I g v E
g m v v E g w v E
ε
τ∞
∞
=
−=
= − −− − − −
• Let ε << 1, use singular perturbation theory
• Define sets of slow and fast equations to govern flow on slow manifold and the fast transitions between the branches of the slow manifold
• In the absence of input, F is boring – interaction between F and synaptic input from O is what will make the problem interesting.
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Effect of O’s inhibition on F
• Inhibition lowers the V-nullcline
Inhibition
( , ) ( - )
( )( )
0 w h e n O i s in a c t iv e1 w h e n O i s a c t iv e
s y ns y n
w
d v f v w g s v Ed t
w v wd wd t v
ss
ε
τ∞
= −
−=
==
S=0
S=1
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Effect of A-current on F( , ) ( ) [ ]
( ) ( )
1 if
if
A K
w
hhl
hhm
dv f v w g n v h v Edt
w v wdwdt v
h v vdh
hdt v v
ε
τ
τ
τ
∞
∞
= − −
−=
−⎧ <⎪⎪= ⎨ −⎪ >⎪⎩
•Low voltage activation & inactivation thresholds v=vh
• Creates a new stable middle branch
• h variable decays along middle and right branches
v=vh
• h and w are slow, v is fast
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Effect of A-current and O’s inhibition on F
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w-h slow manifold
0 ( , ) ( ) [ ]( )
A K
wm
hm
f v w g n v h v Ew v wdw
dtdh hdt
τ
τ
∞
∞
= − −−=
−=
Positively invariant relative to the slow manifold
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Trajectories on w-h slow manifold
• Parameters determine location of the fixed point curve
• w and h time constants determine flow on slow manifold
• Exit through UK implies return to LB, exit through LK implies jump to RB
• When UK, LK, FP all intersect at one point, then singular flow is inconclusive (f), (g)
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n:m periodic solutions
• By changing time constant for h on MB and/or maximal conductance gA of A-current, trajectory can spend more time on MB
• Here the trajectory spends a full O cycle on MB
• We call this a 2:1 solution, 2 cycles of O for 1 cycle of F
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Deriving a one-dimensional map• Variables to start with (v, w, h, s, vO) (5)• Singular perturbation takes care of v (4)• s is slaved to vO (3)• on MB, assume w dynamics faster than h dynamics (2)• Record the value of h every time inhibition is removed,
i.e. when s changes from 1 to 0 (1)• Plus a few more small assumptions yields the one-
dimensional map.1
1
1 11 [ exp( ( ) ) 1]exp( ) if (a)
exp( ) if (b)
n n nin actm m in
hh hh hm hln
n nm in
hm
T Th t t Th
Ph t T
τ τ τ τ
τ
−
−
⎧ + − + − − − <⎪⎪= ⎨⎪ − >⎪⎩
1
[ ]ln
( , )
nn A K
m hmFP
g h v Etf v w
θ
θ
τ− −= (c)
tmn is the time on MB in the nth cycle
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Dynamics of the map• For different values of gA, we obtain different solutions• Discontinuity moves left as gA is increased • Global bifurcation when fixed point disappears• One-dimensional map accurately captures dynamics of
full system
2:1 (gA= 8) 3:2 (gA= 5)
1:1 (gA= 4) 3:1 (gA= 20)
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Iterates of the mapgA = 6.0Period 2
gA = 5.7Period 2
There must be something other than period 1 and 2!
gA = 5.0No Period 2
Bifurcations arise when fixed points of iterates disappear
gA = 5.0Period 3
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The case gA = 5.53
No Period 4No period 1, 2, 3
Yes Period 5
How many different solutions are there?
Can we predict what kinds of solutions will arise?
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Bifurcation diagram• Reveals a rich variety of n:m periodic solutions
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Generalized Pascal Triangle• Farey Rule
• Between each n:1 interval lies a countable infinity of periodic orbits
• Related to border collision bifurcations (Yorke et al).
• Biological relevance?
: : ( ) : ( )a b c d a c b d⊕ = + +
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Remarks• Interaction between A-current of F cell with the
synaptic current from O responsible for interesting dynamics.
• Simple feed-forward model gave rise to a one-dimensional map with rich bifurcation structure, that in some parameter regimes described the dynamics of full five-dimensional system.
• Geometric singular perturbation theory and dynamical systems techniques ideal for studying small neuronal networks.
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Conclusion• Plasticity expands the dynamic capabilities of a network.• In particular, it allows different parameters to control the
network at different frequencies or at different moments in the cycle.
• How plasticity is used depends both on network architecture and on intrinsic properties of neurons.
• Mathematical modeling and analysis can provide insights that are not readily seen from experiments alone.
• Modeling also gives rise to interesting mathematical problems
• On-going work on a variety of related problems.• Work supported by National Science Foundation (US)
and Fulbright-Nehru Program (US & India)