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From Single to Ensemble –
neural synchronization
清華大學腦科學中心
張修明 博士
2005/10/14
They all start from a single neuron
Time scale
Spike
~ms
Brain wave
~10 ms- ~100 ms
LTP
~ hours
Learning
~ minutes- ~hours
Memory
~ minutes- ~years
Initiation
Forming
Maintain
Transfer
Retrieve
How the connection of these neurons can result in something meaningful ?
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Neural connections bring in information
Information
Spatial
spikes@positions
recurrent network
Temporal
Rate--spikes/time
Temporal –-spikes@time
Normal brain waves-- stable persistent phenomena (neural connections)
Normal behavior
Induced “brain waves”
Working memory circuit in the frontal cortex
Epilepsy
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Can inhibitory synapses generate synchronous activity
A simple though nonrealistic system shows it can.
Only one type of ion channel with inactivation process is needed
Oscillation in an inhibitory network
CdV/dt = -gCm3 (V)hi(Vi-VC)-gl(Vi-VL)-gsynsji(Vi-Vsyn)
dh/dt = f[h (Vi)-hi]/th(Vi)
sji = s (Vj) = 1/{1+exp[-(Vj-qsyn)/ksyn]} is instantaneous
with ksyn = 2, gsyn = 0.3 mS/cm2,
VC = 120 mv, VL = -60 mV and
Vsyn = -80 mV -- inhibitory
gL = 0.1 mS/cm2
m (V) = 1/{1 + exp[-(V + 65)/7.8]},
h (V) = 1/{1 + exp[(V + 81)/11]},
th(Vi) = h (V)exp[(V+162.3/17.8)], and
f=3
Wang & Rinzel, 1992 Neural Computation, 4:84
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Two inhibitory neurons can “trigger” each other
resulting in synchronization
No inhibitory signal is transmitted when
V < -44 mV
Set dV/dt = 0 and dh/dt = 0
For gC
= 0.3 mS/cm2
h = [gL(V-VL)+gsyn(V-Vsyn)]/
[gCm3 (V) (VC-V)]
or h = h (V),
gsyn = 0, when no inhibition from the synapse
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Wang & Rinzel, 1992 Neural Computation, 4:84
An activated V1 can not
inhibit V2, if gC is high
enough.
For gC
= 1.0 mS/cm2
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A bistable system can
be triggered into an
oscillation with even
larger gC.
For gC
= 1.5 mS/cm2
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The synchronization can be
in phase or out of phase.
For
dsij/dt = s (Vi)(1-sij)-krsij
and kr is small enough.
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Gamma wave (~40–100 Hz)
The wave is easily observed in EEG on awake animals,
has been suggested to be related to various
daily work, speaking, attention and learing (Miltner et al, 1999, Nature 397:434).
An interneuronal network can generate a coherent oscillatory output to
the pyramidal neurons, thereby providing a substrate for the synaptic
organization of coherent gamma population oscillations.
When metabotropic glutamate receptors were activated, transient oscillatory
IPSPs in the 40 Hz frequency range were observed in pyramidal cells, without
AMPA and NMDA activity. (Whittington et al., 1995, Naute, 373:612)
The interneuron with GABAergic synapses in the hippocampus has been
shown to fire with gamma frequency (Sik et al, 1995, J. Neuroscience:
15:6651).
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Voltage-versus-depth profile of hippocampal field activity in the mouse.
1mv
g
(cx: neocortex; or: stratum oriens; pyr: pyramidal layer; rad: stratum radiatum;
hf: hippocampal fissure: hil, hilus)
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q
G. Buzsaki et al. / Neuroscience 116 (2003) 201–211
Basket cell
interneurons
bouton
Parvalbumin immunoreactive
basket cell and interneurons
in rat hippocampus
o, stratum oriens; p, CA1 pyramidal layer;
r, stratum radiatum.
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Sik et al, 1995, J. Neuroscience: 15:6651
Number of synapse formed by interneurons can be counted
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
Model neuron Each interneuron is described by :
Cm (dV/dt) = -INa - IK - IL - Isyn + Iapp ,
where Cm = 1 mF/cm2 and Iapp is the injected current (in mA/cm2). The leak current
IL = gL(V - EL) has a conductance gL = 0.1 mS/cm2, so that
the passive time constant t0 = Cm/gL = 10 msec; EL = -65 mV.
The spike-generating Na+ and K+ voltage-dependent ion currents (INa
and IK) are of the Hodgkin–Huxley type (Hodgkin and Huxley, 1952).
The transient sodium current
INa = gNam3 h(V - ENa), where the activation variable m is assumed fast
and substituted by its steady-state function
m = am/(am + bm);
am(V) = -0.1(V + 35)/(exp(-0.1(V +35)) - 1),
bm(V) = 4exp(-(V + 60)/18).
The inactivation variable h obeys a first-order kinetics:
dh/dt = f(ah(1 – h) - bhh) where
ah(V) = 0.07 exp(-(V + 58)/20) and
bh(V) = 1/(exp(-0.1(V -28)) + 1).
gNa = 35 mS/cm2;
ENa = 55 mV,
f = 5.
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
The delayed rectifier IK = gKn4 (V - EK), where
the activation variable n obeys the following equation:
dn/dt = f( an(1 – n) - bnn) with
an(V) = -0.01(V + 34)/(exp(-0.1(V + 34)) - 1) and
bn(V) = 0.1-5exp(2(V + 44)/80);
gK = 9 mS/cm2, and
EK = -90 mV.
Parameters are chosen such that the repolarization is 15 mV below the
threshold (~-55 mV) but above the EK and the firing frequency can reach high value.
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
Model synapse. The synaptic current Isyn = gsyns(V - Esyn), where gsyn
is the maximal synaptic conductance and Esyn is the reversal potential.
Typically, we set gsyn = 0.1 mS/cm2 and Esyn = -75 mV (inhibitory).
The gating variable s represents the fraction of open synaptic ion
channels.
ds/dt = aF(Vpre)(1 – s) - bs, where the normalized concentration of the postsynaptic
transmitter receptor complex, F(Vpre), is assumed to be an instantaneous and
sigmoid function of the presynaptic membrane potential, (Perkel et al.,1981; Wang
and Rinzel 1993):
F(Vpre) = 1/(1 + exp(-(Vpre - qsyn)/2)), where
qsyn (set to 0 mV) is high enough so that the transmitter release occurs only when
the presynaptic cell emits a spike.
The channel opening rate a= 12 /msec assures a fast rise of the Isyn, and
the channel closing rate b is the inverse of the decay time constant of the Isyn;
typically, we set b = 0.1/ msec ( tsyn = 10 msec).
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
Random network connectivity. The network model consists of N cells.
The coupling between neurons is randomly assigned, with a fixed average number
of synaptic inputs per neuron, Msyn.
The coherence between two neurons i and j is measured by their cross-correlation
of spike trains at zero time lag within a time bin of Dt = t. More specifically, suppose
that a long time interval T is divided into small bins of t and that two spike trains are
given by X(l) = 0 or 1, Y(l) = 0 or 1, l=1, 2, . . . , K (T/K = t).
Define a coherence measure for the pair as:
kij(t) = SlX(l)Y(l)/[SlX(l)SlY(l)]1/2
The population coherence measure k(t) is defined by the average of kij(t)
over many pairs of neurons in the network. k(t) is between 0 and 1 for all t. For
very small t, k(t) is close to 1 (0) in the case of maximal synchrony (asynchrony).
Initially, the membrane potential is uniformly distributed between -70 and -50 mV
and the other channel-gating variables are set at their corresponding steady-state
values. Coherence was calculated after 1000 msec transients. N = 100 neurons.
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The simulated single interneuron has the
typical excitable and inhibitory property
An increased injection current
causes higher firing frequency
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
A Network coupled by GABAA
synapses can synchronize
Neurons are identical and coupled
in an all-to-all fashion.
The network takes longer time
to synchronize when
kinetics of the Na, K current
slows down (smaller f).
The network results in a twostate synchronization if the
kinetics of the Na, K current
further slows down.
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
No synchronization occurs in the network if the synapse is
excitatory (Esynp = 0 mV), even though each neuron has
more or less the same firing frequency (~43Hz here).
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
The synchronization breaks down if the Iapp is not homogeneous to all neurons
The coherence of the network is reduced
when the standard deviation of the Iapp is
increased, assuming a Gaussian distribution,
although the mean firing frequency does not
change many.
Iapp = 1 mA/cm2
0.03
0.1
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
The number of synapse on each neuron is critical of the network
Synchronization (simulation method ???)
Only ~5 is necessary of synchronization if every neuron
has the same number of synapse, minimum ~40 if
randomly connected in a network of 100 neurons.
The critical synapse number is not sensitive
to the strength of the synapse.
The critical number is increased if the Iapp
intensity is increased.
The size of the network has little effect on
critical synapse number of synchronization.
Implication on epilepsy ?
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
There is an optimal
synaptic time constant for
the coherence of a not-all-connected
and inhomogeneous network.
(Msyn = 60, Is = 0.3)
The optimal synaptic time
constant is about 0.2 of the
mean oscillation period.
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
There are also an optimal
Iapp and synaptic conductance
for the coherence.
In combination, there is an optimal
coherence frequency for this
inhibitory network.
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Wang & Buzsaki, 1996, J. Neurosci.16:6402–6413
Theta oscillations (6–10 Hz), although no consensus has yet emerged,
duringREM sleep (Jouvet, 1969) and
during various activities described by the subjective terms “voluntary,”
“preparatory,” “orienting,” or “exploratory” (Vanderwolf, 1969).
also thought to occur during navigation (Kahana et al, 1999, Nature 399:781)
Theta oscillation is observed in the str. lacunosum-moleculare of
hippocampal CA1 or CA3 region and many other part in the brain such as
entorhinal cortex, amigdala etc.. (Buzsaki, 2002, Neuron 33:325)
Simulation of network synchronization is done for both an isolated
population of medial septal (MS) GABAergic cells and for a reciprocally
inhibitory loop between the MS and the hippocampus.
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Hendelman 2000, Atlas of Functional Neuroanatomy, p189
Medial septal neuron model
CmdV/dt = -INa - IK - IKS - IL - Isyn + I + eh(t)
Horizontal O/A interneurons in Hippocampus (in stratum oriens-alveus)
CmdV/dt = -INa - IK - Ih - ICa - IKCa - IL - Isyn + I
The network is thought to be all-to-all
N = 400
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The presence of KS channel
modulates the firing pattern in
single septal GABAergic neurons
KS activation parameter
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KS affects only the low frequency
interburst firing not the intraburst
activity.
Wang, 2002, J Neurophysiol 87: 889–900
Synchronization at the theta wave range is produced in neither
the septal network nor the hippocampal network.
Synchronization at the gamma
frequency range is possible in
septal region
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No synchronization at the gamma
frequency range is possible in
hippocampus
Wang, 2002, J Neurophysiol 87: 889–900
A coupled septal and hippocampal networks can synchronously
fire at both gamma and theta frequency
Individual
firing is out
of phase
cross the
network
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Wang, 2002, J Neurophysiol 87: 889–900
Only change the synaptic coupling within the septal network
significantly affect the theta frequency.
The pace- maker is
located within
the septal region,
probably according
to the KS channel
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Wang, 2002, J Neurophysiol 87: 889–900
The final theta frequency is stable regardless the different
firing frequencies within the hippocampal network.
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Wang, 2002, J Neurophysiol 87: 889–900
Summary
A stable synchronized firing can occur among inhibitory
neurons.
Individual channel kinetics (decay time etc) may be a major
factor regulating the collective properties of a neural network.
Dynamic mutual interactions generate new properties
beyond the scale of individual elements
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Some thoughts
Do the networks in the brain work separately ?
How sij will become when LTP is considered ?
How a transient synchronization of a neural network form and fall ?
How far away can the “signal” be transmitted from the septal region ?
A better experimental protocol become necessary to investigate the
properties of neurons from individual cells to the whole network
simultaneously.
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