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Effect of Synaptic Plasticity on Brain Rhythms, Associated with Diverse Brain Functions and Diseases, in Complex Biological Neural Networks
(1) Effect of Intrapopulation SpikeTimingDependent Plasticity on Synchronized Rhythms in Complex Neural Networks
We consider a complex neural network consisting of excitatory (E) or inhibitory (I) spiking or bursting neurons.
This neuronal network has adaptive dynamic E to E or I to I intrapopulation synaptic strengths governed by the spiketimingdependent plasticity (STDP).
In previous works without STDP, synchronized population rhythm was found to occur in a range of intermediate noise intensities.
We investigate the effect of additive STDP on the synchronized rhythm by varying the noise intensity. Occurrence of a "Matthew" effect
in synaptic plasticity is found due to a positive feedback process. As a result, good synchronization gets better via longterm potentiation (LTP)
of synaptic strengths, while bad synchronization gets worse via longterm depression (LTD). Furthermore, emergences of longterm potentiation
and longterm depression of synaptic strengths are intensively investigated via microscopic studies based on the distributions of time delays between the pre and the postsynaptic spike times.
[1] S.Y. Kim and W. Lim, "Stochastic spike synchronization in a smallworld neural network with spiketimingdependent plasticity," Neural Networks 97, 92106 (2018).
[2] S.Y. Kim and W. Lim, "Effect of spiketimingdependent plasticity on stochastic burst synchronization in a scalefree neuronal network," Cognitive Neurodynamics 12, 315342 (2018).
[3] S.Y. Kim and W. Lim, "Effect of inhibitory spiketimingdependent plasticity on fast sparsely synchronized rhythms in a smallworld neuronal network," Neural Networks 106, 5066 (2018).
[4] S.Y. Kim and W. Lim, "Burst synchronization in a scalefree neuronal network with inhibitory spiketimingdependent plasticity," Cognitive Neurodynamics 13, 5373 (2019).
(2) Effect of Interpopulation SpikeTimingDependent Plasticity on Synchronized Rhythms in Neuronal Networks with Inhibitory and Excitatory Populations
We consider a twopopulation network consisting of both inhibitory (I) interneurons and excitatory (E) pyramidal cells. This IE neuronal network has adaptive dynamic I to E and E to I interpopulation
synaptic strengths, governed by interpopulation spiketimingdependent plasticity (STDP). In previous works without STDPs, fast sparsely synchronized rhythms, related to diverse cognitive functions,
were found to appear in a range of noise intensity D for static synaptic strengths. Here, by varying D, we investigate the effect of interpopulation STDPs on fast sparsely synchronized rhythms
that emerge in both the I and the Epopulations. Depending on values of D, longterm potentiation (LTP) and longterm depression (LTD) for populationaveraged values of saturated interpopulation synaptic strengths
are found to occur. Then, the degree of fast sparse synchronization varies due to effects of LTP and LTD. In a broad region of intermediate D, the degree of good synchronization (with higher synchronization degree)
becomes decreased, while in a region of large D, the degree of bad synchronization (with lower synchronization degree) gets increased. Consequently, in each I or Epopulation, the synchronization degree becomes
nearly the same in a wide range of D (including both the intermediate and the large D regions). This kind of "equalization effect" is found to occur via cooperative interplay between the average
occupation and pacing degrees of spikes (i.e., the average fraction of firing neurons and the average degree of phase coherence between spikes in each synchronized stripe of spikes in the raster plot of spikes)
in fast sparsely synchronized rhythms. Finally, emergences of LTP and LTD of interpopulation synaptic strengths (leading to occurrence of equalization effect) are intensively investigated via a microscopic method
based on the distributions of time delays between the pre and the postsynaptic spike times.
[1] S.Y. Kim and W. Lim, "Effect of interpopulation spiketimingdependent plasticity on synchronized rhythms in neuronal networks with inhibitory and excitatory populations," Cognitive Neurodynamics 14, 535567 (2020).
(3) Effect of Diverse Recoding of Granule Cells on Optokinetic Response and Delay Eyeblink Conditioning in A Cerebellar Ring Network with Synaptic Plasticity
We develop a cerebellar ring network, and investigate the effect of diverse recoding of granule (GR) cells on optokinetic response (OKR associated with spatial motor control) and delay Pavlovian eyeblink conditioning
(EBC related to temporal motor control) by varying the connection probability p_{c} from Golgi to GR cells. For an optimal value of p_{c}, individual GR cells exhibit diverse spiking patterns. In the case of OKR, they are
inphase, antiphase, or complex outofphase with respect to their population averaged firing activity. In the case of EBC, individual granule cells show various
well and illmatched firing patterns relative to the unconditioned stimulus. Then, these diverselyrecoded signals are fed into the Purkinje cells (PCs) through parallelfibers (PFs),
and the instructor climbingfiber (CF) signals from the inferior olive depress them effectively.
For the OKR, synaptic weights at inphase PFPurkinje cell (PC) synapses of active GR cells are strongly depressed via strong longterm depression (LTD), while those at
antiphase and complex outofphase PFPC synapses are weakly depressed through weak LTD. Similar synaptic plasticity also occurs for the EBC.
In the case of wellmatched PFPC synapses, their synaptic weights are strongly depressed through strong longterm depression (LTD). On the other hand, practically no LTD occurs
for the illmatched PFPC synapses. This kind of effective depression (i.e., strong/weak LTD) at the PFPC synapses causes a big modulation in firings of PCs.
Then, PCs exert effective inhibitory coordination on the vestibular nucleus (VN) neuron (which evokes OKR) doe the OKR and
cerebellar nucleus neuron [which elicits conditioned response (CR)] for the eyeblink conditioning.
In the case of OKR, for the firing of the VN neuron, the learning gain degree L_{g}, corresponding to the modulation gain ratio, increases with increasing the learning cycle, and it
saturates. In the case of EBC, when the learning trial passes a threshold, acquisition of CR begins. In this case, the timing degree T_{d} of CR becomes good due to
presence of the illmatched firing group which plays a role of protection barrier for the timing. With further increase in the trial, strength S of CR (corresponding to the amplitude of eyelid closure)
increases due to strong LTD in the wellmatched firing group, while its timing degree T_{d} decreases. In this way, the well and the illmatched firing groups play their own roles
for the strength and the
timing of CR, respectively. Thus, with increasing the learning trial, the (overall) learning efficiency L_{e} (taking into consideration both timing and strength of CR) for the CR is increased, and
eventually it becomes saturated. By changing p_{c} from its optimal value, we also investigate the effect of diverse recoding on the OKR and the EBC. It is thus found that the more diverse
in recoding of GR cells, the more effective in motor learning for the OKR and the Pavlovian EBC.
[1] S.Y. Kim and W. Lim, "Effect of diverse recoding of granule cells on optokinetic response in a cerebellar ring network with synaptic plasticity," eprint: arXiv:2003.11325[qbio.NC]; bioRxiv: DOI:10.1101/2020.03.25.007245.
[2]
S.Y. Kim and W. Lim, "Influence of various temporal recoding on Pavlovian eyeblink conditioning in the cerebellum," eprint: arXiv:2006.14933[qbio.NC]; bioRxiv: DOI:10.1101/2020.06.23.168294.
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Sparsely
Synchronized Brain Rhythms in Complex Neural Networks
Sparselysynchronized brain rhythms,
associated with diverse cognitive functions such as sensory perception, feature
integration, selective attention, and memory formation, have been observed in
electric recordings (e.g., EEG and local field potentials) of brain activity. At
the population level, cortical rhythms exhibit smallamplitude fast
oscillations, while at the cellular level individual neurons show stochastic
firings sparsely at a much lower rate than the population rate. To resolve the
apparent dichotomy between synchrony at the population level and stochasticity
at the cellular level, we study emergence of sparsely synchronized brain rhythms
in networks of Geigercounterlike neurons exhibiting irregular discharges.
Previously, sparse synchronization was found to occur for cases of both global
coupling (i.e., regular alltoall coupling) and random coupling. However, a
real neural network is known to be nonregular and nonrandom because it has
complex topology (e.g., smallworldness, scalefreeness, and modularity). Hence,
we investigate the effect of the network architecture on emergence of sparsely
synchronized rhythms in real complex networks such as the smallworld, the
scalefree, the starlike, and the clustered networks.
(1) Sparsely Synchronized Brain Rhythm in A SmallWorld
Neural Network
Sparsely synchronized cortical
rhythms, associated with diverse cognitive functions, have been
observed in electric recordings of brain activity. At the population
level, cortical rhythms exhibit smallamplitude fast oscillations,
while at the cellular level, individual neurons show stochastic
firings sparsely at a much lower rate than the population rate. We
study the effect of network architecture on sparse synchronization
in an inhibitory population of Geigercounterlike neurons showing
stochastic discharges. As an element of the network, we consider the
subthreshold MorrisLecar neuron and the fast spiking Izhikevich
interneuron which fires as Geiger counters. When strong external
noise balances with strong synaptic inhibition, sparsely
synchronized rhythms are found to emerge. Previously, sparse
synchronization was found to occur for both cases of global coupling
(i.e., regular alltoall coupling) and random coupling. However, a
real neural network is known to be neither regular, nor random.
Here, we consider sparse WattsStrogatz smallworld networks which
interpolates between the regular lattice and the random graph via
rewiring. We start from the regular lattice with only shortrange
connections, and then investigate emergence of sparse
synchronization by increasing the rewiring probability p from
shortrange to longrange connections. For p=0, the average synaptic
path length between pairs of neurons becomes long, and hence there
exists only an unsynchronized population state because the global
efficiency of information transfer is low. However, as p is
increased, longrange connections begin to appear, and global
effective communication between distant neurons may be available via
shorter synaptic paths. Consequently, as p passes a threshold p_{th},
sparsely synchronized population rhythms emerge. However, with
increasing p longer axon wirings become expensive because of their
material and energy costs. At an optimal value p^{*}_{DE}
of the rewiring probability, the ratio of the synchrony degree to
the wiring cost is found to become maximal. In this way, an optimal
sparse synchronization is found to occur at a minimal wiring cost in
an economic smallworld network through tradeoff between synchrony
and wiring cost.
[1] S.Y. Kim and W. Lim,
"Sparselysynchronized brain rhythm in a smallworld neural network"
J. Korean Phys. Soc. 63, 104113 (2013). [2]
S.Y. Kim and W. Lim,
"Effect of smallworld connectivity on fast sparsely
synchronized cortical rhythms,"
Physica A 421, 109123 (2015).
(2) Sparsely Synchronized Brain Rhythm in A ScaleFree
Neural Network
We consider a directed version of
the BarabasiAlbert scalefree network model with symmetric
preferential attachment with the same in and outdegrees and study
the emergence of sparsely synchronized rhythms for a fixed
attachment degree in an inhibitory population of fastspiking
Izhikevich interneurons. Fast sparsely synchronized rhythms with
stochastic and intermittent neuronal discharges are found to appear
for large values of J (synaptic inhibition strength) and D (noise
intensity). For an intensive study we fix J at a sufficiently large
value and investigate the population states by increasing D. For
small D, full synchronization with the same populationrhythm
frequency f_{P} and mean firing rate (MFR) f_{i} of
individual neurons occurs, while for large D partial synchronization
with f_{P} > <f_{i}> (<f_{i}>:
ensembleaveraged MFR) appears due to intermittent discharge of
individual neurons; in particular, the case of f_{P} > 4<f_{i}>
is referred to as sparse synchronization. For the case of partial
and sparse synchronization, MFRs of individual neurons vary
depending on their degrees. As D passes a critical value D^{*}
(which is determined by employing an order parameter), a transition
to unsynchronization occurs due to the destructive role of noise to
spoil the pacing between sparse spikes. For D<D^{*},
population synchronization emerges in the whole population because
the spatial correlation length between the neuronal pairs covers the
whole system. Furthermore, the degree of population synchronization
is also measured in terms of two types of realistic
statisticalmechanical measures. Only for the partial and sparse
synchronization do contributions of individual neuronal dynamics to
population synchronization change depending on their degrees, unlike
in the case of full synchronization. Consequently, dynamics of
individual neurons reveal the inhomogeneous network structure for
the case of partial and sparse synchronization, which is in contrast
to the case of statistically homogeneous random graphs and
smallworld networks. Finally, we investigate the effect of network
architecture on sparse synchronization for fixed values of J and D
in the following three cases: (1) variation in the degree of
symmetric attachment, (2) asymmetric preferential attachment of new
nodes with different in and outdegrees, and (3) preferential
attachment between preexisting nodes (without addition of new
nodes). In these three cases, both relation between network topology
(e.g., average path length and betweenness centralization) and
sparse synchronization and contributions of individual dynamics to
the sparse synchronization are discussed.
[1] S.Y. Kim and W. Lim, "Fast sparsely
synchronized brain rhythms in a scalefree neural network," Phys.
Rev. E 92, 022717 (2015).
(3) Sparsely Synchronized Brain Rhythm in Clustered
SmallWorld Neural Networks
We consider a clustered network
with smallworld subnetworks of inhibitory fast spiking
interneurons, and investigate the effect of intermodular connection
on emergence of fast sparsely synchronized rhythms by varying both
the intermodular coupling strength J_{inter} and the
average number of intermodular links per interneuron M^{(inter)}_{syn}.
In contrast to the case of nonclustered networks, two kinds of
sparsely synchronized states such as modular and global
synchronization are found. For the case of modular sparse
synchronization, the population behavior reveals the modular
structure, because the intramodular dynamics of subnetworks make
some mismatching. On the other hand, in the case of global sparse
synchronization, the population behavior is globally identical,
independently of the cluster structure, because the intramodular
dynamics of subnetworks make perfect matching. We introduce a
realistic crosscorrelation modularity measure, representing the
matchingdegree between the instantaneous subpopulation spike rates
of the subnetworks, and examine whether the sparse synchronization
is global or modular. Depending on its magnitude, the intermodular
coupling strength J_{inter} seems to play ¡°dual¡± roles for
the pacing between spikes in each subnetwork. For large J_{inter},
due to strong inhibition it plays a destructive role to ¡°spoil¡± the
pacing between spikes, while for small J_{inter} it plays a
constructive role to ¡°favor¡± the pacing between spikes. Through
competition between the constructive and the destructive roles of J_{inter},
there exists an intermediate optimal J_{inter} at which the
pacing degree between spikes becomes maximal. In contrast, the
average number of intermodular links per interneuron M^{(inter)}_{syn}
seems to play a role just to favor the pacing between spikes. With
increasing M^{(inter)}_{syn}, the pacing degree
between spikes increases monotonically thanks to the increase in the
degree of effectiveness of global communication between spikes.
Furthermore, we employ the realistic sub and wholepopulation order
parameters, based on the instantaneous sub and wholepopulation
spike rates, to determine the threshold values for the
synchronizationunsynchronization transition in the sub and
wholepopulations, and the degrees of global and modular sparse
synchronization are also measured in terms of the realistic sub and
wholepopulation statisticalmechanical spiking measures defined by
considering both the occupation and the pacing degrees of spikes. It
is expected that our results could have implications for the role of
the brain plasticity in some functional behaviors associated with
population synchronization.
[1] S.Y. Kim and W. Lim, "Effect of intermodular
connection on fast sparse synchronization in clustered smallworld neural
networks," Phys. Rev. E 92, 052716 (2015).
Burst Synchronization in Complex Neural Systems
We are concerned about
population synchronization of bursting neurons. Bursting occurs when
neuronal activity alternates, on a slow timescale, between a silent
phase and an active (bursting) phase of fast repetitive spikings. This
type of bursting activity occurs due to the interplay of the fast ionic
currents leading to spiking activity and the slower currents modulating
the spiking activity. Hence, the dynamics of bursting neurons have two
timescales: slow bursting timescale and fast spiking timescale. These
bursting neurons exhibit two different patterns of synchronization due
to the slow and the fast timescales of bursting activity. Burst
synchronization (synchrony on the slow bursting timescale) refers to a
temporal coherence between the active phase (bursting) onset or offset
times of bursting neurons, while spike synchronization (synchrony on the
fast spike timescale) characterizes a temporal coherence between
intraburst spikes fired by bursting neurons in their respective active
phases. We study burst and spike synchronization of bursting neurons,
associated with neural information processes in health and disease, in
complex networks such as smallworld, scalefree, and clustered
networks.
(1)
Stochastic Bursting Synchronization in A Population of Subthreshold
Izhikevich Neurons
We
are interested in neural bursting activity (alternating between a
silent phase and an active phase of repetitive spiking). Cortical
intrinsically bursting neurons, thalamocortical relay neurons,
thalamic reticular neurons, and hippocampal pyramidal neurons are
representative examples of bursting neurons. We investigate coherent
population dynamics in these bursting neurons by varying the noise
intensity D. Such coherence is well visualized in the raster plot of
neural spikings. For a coherent state, burst bands, composed of
stripes of spikes, constitute the raster plot. For this case, burst
synchronization refers to a temporal relationship between active
phase onset or offset times of bursting neurons, while spike
synchronization characterizes a temporal relationship between spikes
fired by different bursting neurons in their respective active
phases. For the coherent case, in addition to burst synchronization,
spike synchronization also occurs in each burst band. As a result of
this complete synchronization, the (ensembleaveraged) global
membrane potential V_{G} exhibits a bursting activity like the local
membrane potential (i.e., fast spikes appear on a slow wave in V_{G}).
However, as D is increased, loss of spike coherence first occurs in
each burst band due to smearing of stripes of spikes. With further
increase in D, overlapping between bands begins to occur, which
eventually leads to complete loss of burst synchronization. To
characterize these burst and spike coherence, we introduce a new
type of coherence measure quantifying the degree of coherence seen
in the raster plot of neural spiking, where the global potential is
used to give a reference phase for the burst onset times and the
spiking times in active phases.
[1]
S.Y. Kim, Y. Kim, D.G. Hong, J. Kim, and W. Lim, "Stochastic
bursting synchronization in a population of subthreshold Izhikevich
neurons", J. Korean Phys. Soc. 60, 14411447 (2012).
(2)
CouplingInduced Population Synchronization in An Excitatory Population
of Subthreshold Izhikevich Neurons
We
consider an excitatory population of subthreshold Izhikevich neurons
which exhibit noiseinduced firings. By varying the coupling
strength J, we investigate population synchronization between the
noiseinduced firings which may be used for efficient cognitive
processing such as sensory perception, multisensory binding,
selective attention, and memory formation. As J is increased, rich
types of population synchronization (e.g., spike, burst, and fast
spike synchronization) are found to occur. Transitions between
population synchronization and incoherence are well described in
terms of an order parameter O. As a final step, the coupling induces
oscillator death (quenching of noiseinduced spikings) because each
neuron is attracted to a noisy equilibrium state. The oscillator
death leads to a transition from firing to nonfiring states at the
population level, which may be well described in terms of the
timeaveraged population spike rate R. In addition to the
statistical mechanical analysis using O and R, each population and
individual state are also characterized by using the techniques of
nonlinear dynamics such as the raster plot of neural spikes, the
time series of the membrane potential, and the phase portrait. We
note that population synchronization of noiseinduced firings may
lead to emergence of synchronous brain rhythms in a noisy
environment, associated with diverse cognitive functions.
[1] S.Y. Kim and W. Lim,
"Couplinginduced population synchronization in an excitatory
population of subthreshold Izhikevich neurons,"
Cognitive Neurodynamics 7, 495503 (2013).
(3)
NoiseInduced Burst And Spike Synchronizations in An Inhibitory
SmallWorld Network of Subthreshold Bursting Neurons
We
are interested in noiseinduced firings of subthreshold neurons
which may be used for encoding environmental stimuli. Noiseinduced
population synchronization was previously studied only for the case
of global coupling, unlike the case of subthreshold spiking neurons.
Hence, we investigate the effect of complex network architecture on
noiseinduced synchronization in an inhibitory population of
subthreshold bursting Hindmarsh–Rose neurons. For modeling complex
synaptic connectivity, we consider the Watts–Strogatz smallworld
network which interpolates between regular lattice and random
network via rewiring, and investigate the effect of smallworld
connectivity on emergence of noiseinduced population
synchronization. Thus, noiseinduced burst synchronization
(synchrony on the slow bursting time scale) and spike
synchronization (synchrony on the fast spike time scale) are found
to appear in a synchronized region of the J–D plane (J: synaptic
inhibition strength and D: noise intensity). As the rewiring
probability p is decreased from 1 (random network) to 0 (regular
lattice), the region of spike synchronization shrinks rapidly in the
J–D plane, while the region of the burst synchronization decreases
slowly. We separate the slow bursting and the fast spiking time
scales via frequency filtering, and characterize the noiseinduced
burst and spike synchronizations by employing realistic order
parameters and statisticalmechanical measures introduced in our
recent work. Thus, the bursting and spiking thresholds for the burst
and spike synchronization transitions are determined in terms of the
bursting and spiking order parameters, respectively. Furthermore, we
also measure the degrees of burst and spike synchronizations in
terms of the statisticalmechanical bursting and spiking measures,
respectively.
[1]
S.Y. Kim and W. Lim,
"Noiseinduced burst and spike synchronizations in an inhibitory
smallworld network of subthreshold bursting neurons," Cognitive
Neurodynamics 9, 179200 (2015).
(4)
Effect of Network Architecture on Burst and Spike Synchronization in A
ScaleFree Network of Bursting Neurons
We
investigate the effect of network architecture on burst and spike
synchronization in a directed scalefree network (SFN) of bursting
neurons, evolved via two independent ¥á− and ¥â−processes. The
¥á−process corresponds to a directed version of the BarabasiAlbert
SFN model with growth and preferential attachment, while for the
¥â−process only preferential attachments between preexisting nodes
are made without addition of new nodes. We first consider the ¡°pure¡±
¥á−process of symmetric preferential attachment (with the same in
and outdegrees), and study emergence of burst and spike
synchronization by varying the coupling strength J and the noise
intensity D for a fixed attachment degree. Characterizations of
burst and spike synchronization are also made by employing realistic
order parameters and statisticalmechanical measures. Next, we
choose appropriate values of J and D where only the burst
synchronization occurs, and investigate the effect of the scalefree
connectivity on the burst synchronization by varying (1) the
symmetric attachment degree and (2) the asymmetry parameter
(representing deviation from the symmetric case) in the ¥á−process,
and (3) the occurrence probability of the ¥â−process. In all these
three cases, changes in the type and the degree of population
synchronization are studied in connection with the network topology
such as the degree distribution, the average path length L_{p},
and the betweenness centralization B_{c}.
It is thus found that not only L_{p}
and B_{c}
(affecting global communication between nodes) but also the
indegree distribution (affecting individual dynamics) are important
network factors for effective population synchronization in SFNs.
¡¡
[1] S.Y. Kim and W. Lim, "Effect of
network architecture on burst and spike synchronization in a
scalefree network of bursting neurons," Neural Networks 79, 5377
(2016).
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Thermodynamic and StatisticalMechanical Measures for Characterization
of Neural Synchronization
¡¡
Synchronized brain rhythms in sleep and awake
states (e.g., alpha, sleep spindle, gamma, ultrafast, beta rhythms), associated
with diverse sensory and cognitive functions, have been observed in electrical
recordings (EEG and local field potentials) of brain activity. Neural
synchronization may be well described by using the populationaveraged global
potential V_{G} in computational neuroscience, However, to obtain V_{G}
in real experiments is practically difficult. Instead of V_{G}, the
instantaneous population spike rate (IPSR), which may be obtained
experimentally, is used for description of population behaviors in both
computational and experimental neuroscience. The timeaveraged fluctuation of IPSR
plays the role of an order parameter O used for describing the
synchronyasynchrony transition in neural systems. This order parameter can be
regarded as a thermodynamic measure because it concerns only the macroscopic IPSR
without considering any relation between IPSR and
microscopic individual potentials (spikes). Population spike synchronization may
be well seen in the raster plot of neural spikes. The degree of collective spike
synchronization seen in the raster plot may be well measured in terms of a new
``statisticalmechanical'' spikebased measure M_{s} introduced by
considering the occupation and the pacing patterns of spikes in the raster plot.
In particular, the pacing degree between spikes is determined in a
statisticalmechanical way by quantifying the average contribution of
(microscopic) individual spikes to the macroscopic IPSR.
This ``statisticalmechanical'' measure M_{s} is in contrast to the
conventional measures such as the ``thermodynamic'' order parameter (which
concerns the timeaveraged fluctuations of the macroscopic global potential),
the ``microscopic'' correlationbased measure (based on the crosscorrelation
between the microscopic individual potentials), and the measures of precise
spike timing [based on the peristimulus time histogram (PSTH)]. For the
conventional PSTHbased measure ``events,'' corresponding to peaks of the IPSR,
are selected through setting a threshold. Then, the measures for the reliability
and the precision of spike timing concern only the spikes within the events, in
contrast to the case of the ``statisticalmechanical'' measure where all spikes
are considered (without selecting events). A main difference between the
conventional and the new spikebased measures lies in determining the pacing
degree of spikes. The precision of spike timing for the conventional case is
given by just the standard deviation of (microscopic) individual spike times
within an event without considering the quantitative contribution of
(microscopic) individual spikes to the (macroscopic) global activity. Hence, the
PSTHbased measure is not a statisticalmechanical measure. Finally, we
emphasize that both the realistic thermodynamic and statisticalmechanical
measures, based on the IPSR, may be practically applicable for characterization
of neural synchronization in both the computational and the experimental
neuroscience such as spiketiming reliability, stimulus discrimination,
and event related sync./desync.
(1)
StatisticalMechanical Measure of Stochastic Spiking Coherence in A Population
of Inhibitory Subthreshold Neurons
By
varying the noise intensity, we study stochastic spiking coherence
(i.e., collective coherence between noiseinduced neural spikings)
in an inhibitory population of subthreshold neurons (which cannot
fire spontaneously without noise). This stochastic spiking coherence
may be well visualized in the raster plot of neural spikes. For a
coherent case, partiallyoccupied "stripes" (composed of spikes and
indicating collective coherence) are formed in the raster plot. This
partial occupation occurs due to "stochastic spike skipping" which
is well shown in the multipeaked interspike interval histogram. The
main purpose of our work is to quantitatively measure the degree of
stochastic spiking coherence seen in the raster plot. We introduce a
new spikebased coherence measure M_{s} by considering the
occupation pattern and the pacing pattern of spikes in the stripes.
In particular, the pacing degree between spikes is determined in a
statisticalmechanical way by quantifying the average contribution
of (microscopic) individual spikes to the (macroscopic)
ensembleaveraged global potential. This "statisticalmechanical"
measure M_{s} is in contrast to the conventional measures
such as the "thermodynamic" order parameter (which concerns the
timeaveraged fluctuations of the macroscopic global potential), the
"microscopic" correlationbased measure (based on the
crosscorrelation between the microscopic individual potentials),
and the measures of precise spike timing (based on the peristimulus
time histogram). In terms of M_{s}, we quantitatively
characterize the stochastic spiking coherence, and find that M_{s}
reflects the degree of collective spiking coherence seen in the
raster plot very well. Hence, the "statisticalmechanical"
spikebased measure M_{s} may be used usefully to quantify
the degree of stochastic spiking coherence in a
statisticalmechanical way.
[1] W. Lim and S.Y.
Kim, "StatisiticalMechanical Measure of Stochastic Spiking
Coherence in A Population of Inhibitory Subthreshold Neuron", J.
Comput. Neurosci. 31, 667677 (2011).
[2] S.Y. Kim and W. Lim, "Realistic thermodynamic
and statisticalmechanical measures for neural synchronization," J. Neurosci.
Methods 226, 161170 (2014).
(2)
StatisticalMechanical Measure of Burst and Spike Synchronizations
of Bursting
Neurons
We
are interested in characterization of population synchronization of
bursting neurons which exhibit both the slow bursting and the fast
spiking timescales, in contrast to spiking neurons. Population
synchronization may be well visualized in the raster plot of neural
spikes which can be obtained in experiments. The instantaneous
population firing rate (IPFR), which may be directly obtained from
the raster plot of spikes, is often used as a realistic collective
quantity describing population behaviors in both the computational
and the experimental neuroscience. For the case of spiking neurons,
realistic thermodynamic order parameter and statistical–mechanical
spiking measure, based on IPFR, were introduced in our recent work
to make practical characterization of spike synchronization. Here,
we separate the slow bursting and the fast spiking timescales via
frequency filtering, and extend the thermodynamic order parameter
and the statistical–mechanical measure to the case of bursting
neurons. Consequently, it is shown in explicit examples that both
the order parameters and the statistical–mechanical measures may be
effectively used to characterize the burst and spike
synchronizations of bursting neurons.
[1] S.Y. Kim and W.
Lim, "Thermodynamic order parameters and statisticalmechanical
measures for characterization of the burst and spike
synchronizations of bursting neurons," Physica A 438, 544559
(2015). [2] S.Y. Kim and W. Lim, "Frequencydomain order parameters for the
burst and spike synchronization transitions of bursting neurons,"
Cognitive Neurodynamics 9, 411421 (2015). ¡¡
Stochastic Spiking
Coherence in Networks of Subthreshold Neurons
In recent years,
much attention has been paid to brain rhythms. Synchronization of the
firing activity in groups of neurons may be used for efficient sensory
processing (e.g., visual binding). In addition to a constructive role of
encoding sensory stimuli, neural synchronization is also correlated with
pathological rhythms associated with neural diseases (e.g., epileptic
seizures and tremors in Parkinson's disease). Many studies on collective
dynamical behaviors in neural systems were made to understand the
mechanisms of such synchronized firings. However, most of them were
restricted to the suprathreshold case consisting of spontaneously firing
(i.e., selfoscillating) neurons. We note that in addition to
suprathreshold neurons, subthreshold neurons (which can fire only with
the help of noise) also exist in real neural systems. Hence, we must
take into consideration the existence of subthreshold neurons for the
study on the population neurodynamics. For the subthreshold case,
neurons cannot fire spontaneously without noise. Noise is usually
considered as a nuisance, degrading the performance of dynamical
systems. However, in certain circumstances, noise plays a constructive
role in the emergence of dynamical order. A main subject of our study is
to investigate stochastic spiking coherence (i.e., collective coherence
between noiseinduced firings) for the subthreshold case. Based on this
study for the subthreshold case, we also study coherent population
dynamics in heterogeneous ensembles composed of subthreshold and
suprathreshold neurons.
(1) Stochastic Spiking
Coherence in Coupled Excitatory Neurons
A.
Characterization of Stochastic Spiking Coherence
We consider a large population of subthreshold MorrisLecar neurons interacting via global instantaneous
pulsetype excitatory synaptic coupling. By varying the noise intensity, we
investigate numerically stochastic spiking coherence (i.e., collective coherence
between noiseinduced neural spikings). As the noise amplitude passes a
threshold, a transition from an incoherent to a coherent state occurs. This
coherent transition is described in terms of the "thermodynamic" order
parameter, which concerns a macroscopic timeaveraged fluctuation of the global
potential. We note that such stochastic spiking coherence may be well visualized
in terms of the raster plot of neural spikings (i.e., spatiotemporal plot of
neural spikings), which is directly obtained in experiments. To quantitatively
measure the degree of stochastic spiking coherence (seen in the raster plot), we
introduce a new type of ``spiking coherence measure,'' by taking into
consideration the average contribution of (microscopic) local neural spikings to
the (macroscopic) global membrane potential. Hence, the spiking coherence
measure may be regarded as a "statisticalmechanical" measure. Through
competition between the constructive and the destructive roles of noise,
stochastic spiking coherence is found to occur over a large range of
intermediate noise intensities and to be well characterized in terms of the
mutually complementary quantities of the order parameter and the spiking
measure. Particularly, the spiking measure reflects the degree of stochastic
spiking coherence seen in the raster plot very well.
[1] W. Lim and S.Y. Kim, "Characterization
of Stochastic Spiking Coherence in Coupled Neurons," J. Korean Phys. Soc. 51,
14271431 (2007).
[2] W. Lim and S.Y. Kim, "Stochastic spiking
coherence in coupled subthreshold MorrisLecar neurons,"
Int. J. Mod. Phys. B 23, 703710 (2009).
B. Stochastic
Oscillator Death
We consider an ensemble
of subthreshold MorrisLecar neurons interacting via global instantaneous
pulsetype excitatory synaptic coupling. As the coupling strength passes a lower
threshold, the coupling stimulates coherence between noiseinduced spikings.
This coherent transition is well described in terms of an order parameter.
However, for sufficiently large coupling strength, "stochastic oscillator death"
(i.e., quenching of noiseinduced spikings), leading to breakup of collective
spiking coherence, is found to occur. Using the techniques of nonlinear
dynamics, we investigate the dynamical origin of stochastic oscillator death.
Thus, we show that stochastic oscillator death occurs because each local neuron
is attracted to a noisy equilibrium state via an infiniteperiod bifurcation.
Furthermore, we introduce a new "statisticalmechanical" parameter, called the
average firing probability, and quantitatively characterize a transition from
firing to nonfiring states which results from stochastic oscillator death. For
a firing (nonfiring) state, the averaging firing probability tends to be
nonzero (zero) in the thermodynamic limit. We note that the role of the average
firing probability for the firingnonfiring transition is similar to that of the
order parameter used for the coherenceincoherence transition.
[1] W. Lim and S.Y. Kim, "Stochastic Oscillator
Death in Globally Coupled Neural Systems," J. Korean Phys. Soc. 52, 19131917
(2008).
[2] W. Lim and S.Y. Kim, "Couplinginduced
spiking coherence in coupled subthreshold neurons," Int. J. Mod. Phys. B 23,
21492157 (2009).
(2) Stochastic Spiking
Coherence in Coupled Inhibitory Neurons
Depending
on the type of synaptic receptors (e.g., AMPA and GABA), the synaptic coupling
may be excitatory or inhibitory. About 20 % of neurons are inhibitory ones in
the human brain. For example, the principal neurons in the cortex are excitatory
ones, while the interneurons are inhibitory ones. Through the efficient roles of
the inhibitory neurons, the functions of the excitatory neurons are diversified
and their computational abilities are much enhanced. In such a way, the brain
functions may be well performed via balance of excitation and inhibition. Here, we study coherent population dynamics in coupled
inhibitory neurons, and compare them with those in coupled excitatory neurons.
A.
StatisticalMechanical Measure of Stochastic Spiking Coherence in A
Population of Inhibitory Subthreshold Neurons
By varying the noise intensity, we study stochastic spiking coherence
(i.e., collective coherence between noiseinduced neural spikings) in an inhibitory population of subthreshold neurons (which
cannot fire spontaneously without noise). This stochastic spiking coherence may be well visualized in the raster plot of neural
spikes. For a coherent case, partiallyoccupied "stripes" (composed of spikes and indicating collective coherence) are formed
in the raster plot. This partial occupation occurs due to "stochastic spike skipping" which is well shown in the multipeaked interspike interval histogram. The main purpose of our work is to quantitatively measure the degree of stochastic spiking coherence
seen in the raster plot. We introduce a new spikebased coherence measure
M_{s} by considering the occupation pattern and the
pacing pattern of spikes in the stripes. In particular, the pacing degree between spikes is determined in a statisticalmechanical
way by quantifying the average contribution of (microscopic) individual spikes to the (macroscopic) ensembleaveraged global potential.
This "statisticalmechanical" measure
M_{s} is in contrast to the conventional measures such as the "thermodynamic" order parameter
(which concerns the timeaveraged fluctuations of the macroscopic global potential), the "microscopic" correlationbased measure
(based on the crosscorrelation between the microscopic individual potentials), and the measures of precise spike timing (based on the
peristimulus time histogram). In terms of M_{s}, we quantitatively characterize the stochastic spiking coherence, and find that
M_{s}
reflects the degree of collective spiking coherence seen in the raster plot very well. Hence, the "statisticalmechanical" spikebased
measure M_{s }may be used usefully to quantify the degree of stochastic spiking coherence in a statisticalmechanical way.
[1] W. Lim and S.Y. Kim, "StatisticalMechanical Measure of Stochastic Spiking Coherence in A Population of
Inhibitory Subthreshold Neuron", J. Comput. Neurosci. 31, 667677 (2011).
B. Inhibitory
Coherence in A Heterogeneous Population of Subthreshold and
Suprathreshold TypeI Neurons
We study inhibitory coherence (i.e., collective coherence by synaptic inhibition) in a population of globally coupled typeI neurons which can fire at arbitrarily low frequency. No inhibitory coherence is observed in a homogeneous population composed of only subthreshold neurons which exhibit noiseinduced firings. In addition to subthreshold neurons, there exist spontaneously firing suprathreshold neurons in a noisy environment of a real brain. To take into consideration the effect of suprathreshold neurons on inhibitory coherence, we consider a heterogeneous population of subthreshold and suprathreshold neurons, and investigate the inhibitory coherence by increasing the fraction of suprathreshold neurons
P_{supra}. As
P_{supra} passes a threshold
P_{supra}, suprathreshold neurons begin to synchronize and play the role of coherent inhibitors for the emergence of inhibitory coherence. Thus, regularlyoscillating populationaveraged global potential appears for
P_{supra}>
P^{*}_{supra}. For this coherent case suprathreshold neurons exhibit sparse spike synchronization (i.e., individual potentials of suprathreshold neurons consist of coherent sparse spikings and coherent subthreshold smallamplitude hoppings). By virtue of their coherent inhibition, sparsely synchronized suprathreshold neurons suppress noisy activity of subthreshold neurons. Thus, subthreshold neurons exhibit hopping synchronization (i.e., only coherent subthreshold hopping oscillations without spikings appear in the individual potentials of subthreshold neurons). We also characterize the inhibitory coherence in terms of the "statisticalmechanical" spikebased and correlationbased measures which quantify the average contributions of the microscopic individual spikes and individual potentials to the macroscopic global potential. Finally, effect of sparse randomness of synaptic connectivity on the inhibitory coherence is briefly discussed.
[1] S.Y. Kim, D.G. Hong, J. Kim, and W. Lim, "Inhibitory coherence in a heterogeneous population of subthreshold and suprathreshold typeI neurons", J. Phys. A 45, 155102 (2012).
C. Effect of
Sparse Random Connectivity on the Stochastic Spiking Coherence of
Inhibitory Subthreshold Neurons
We study the effect of network structure on the stochastic spiking coherence (i.e., collective
coherence between noiseinduced neural spikings) in an inhibitory population of subthreshold neurons (which cannot fire spontaneously without noise).
Previously, stochastic spiking coherence was found to occur for the case of global coupling. However, "sparseness" of a real neural network is well known.
Hence, we investigate the effect of sparse random connectivity on the stochastic spiking coherence by varying the average number of synaptic inputs per
neuron M_{syn}. It is thus found that stochastic spiking coherence emerges if
M_{syn} is larger than a (very small) critical value
M^{*}_{syn}, independently
of the network size N. This stochastic spiking coherence may be well visualized in the raster plot of neural spikes. For a coherent case, partiallyoccupied
stripes appear. As M_{syn} is decreased from
N1 (globallycoupled case), the average occupation degree of spikes per stripe increases very slowly.
On the other hand, the average pacing degree between spikes per stripe decreases slowly, but near
M^{*}_{syn} its decrease becomes very rapid. This decrease
in the pacing degree can also be well seen through merging of multiple peaks in the interspike interval histograms. Due to the effect of the pacing degree,
the degree of stochastic spiking coherence decreases abruptly near the threshold
M^{*}_{syn}.
[1] D.G. Hong, S.Y. Kim, and W. Lim, "Effect of Sparse Random Connectivity on the Stochastic Spiking Coherence of Inhibitory Subthreshold Neurons," J. Korean Phys. Soc. 59, 28402846 (2011).
(3) Stochastic Spiking
Coherence in Two Populations of Excitatory and Inhibitory Neurons
Human brain exhibits their
functions well through balance between the excitatory and the
inhibitory neurons. The ratio of inhibitory neurons in the brain is
about 20%. We are interested in collective spiking coherence in two
populations of excitatory and inhibitory neurons. Particularly, we
study the effect of the interpopulation coupling on such spiking coherence. So, by varying the interpopulation coupling strength to keep the E/I balance (i.e., the ratio of AMPA and the GABA conductances is equal for both E and Ineurons), we investigate coherent population dynamics in each population. Such population dynamics is analyzed using the techniques of the nonlinear dynamics and the statistical mechanics. We also introduce the interpopulation spiking measure between the source and the target populations, and characterize the degree of the interpopulation effect.
A. Interpopulation
Effect on Stochastic Spiking Coherence in Two Populations of
Excitatory and Inhibitory Subthreshold TypeI Neurons
We consider two populations of excitatory and inhibitory subthreshold typeI neurons which can fire at arbitrary low frequency. No coherence occurs in the single inhibitory population, while stochastic excitatory coherence appears in the single excitatory population. Through the coupling between the excitatory source population and the inhibitory target population, coherence may appear in the inhibitory population. By varying the interpopulation coupling strength to keep the E/I balance, we investigate the effect of the interpopulation coupling on the stochastic spiking coherence in each population. The population dynamics in each population is analyzed using the techniques of the nonlinear dynamics and the statistical mechanics. Particularly, we introduce the interpopulation spiking measure between the source and the target population, and characterize the degree of the interpopulationcoupling effect.
Strange Nonchaotic Firing in
the Quasiperiodicallyforced Neuron
We
study the transition from a silent state to a spiking state by varying the DC
stimulus in the quasiperiodicallyforced neuron. For this
quasiperiodicallyforced case, a new type of strange nonchaotic (SN) firing
(spiking or bursting) state is found to appear between the silent state and the
chaotic firing state as an intermediate one. Using a rational approximation to
the quasiperiodic forcing, we investigate the mechanism for the appearance of
such an SN firing state. We thus find that a smooth torus (corresponding to the
silent state) is transformed into an SN firing attractor via a phasedependent
bifurcation. This is in contrast to the periodicallyforced case where the
silent state transforms directly to a chaotic spiking state. These SN firing
states are also found to be be aperiodic complex ones, as in the case of chaotic
spiking states. Hence, aperiodic complex spikings may result from two
dynamically different states with strange geometry (one is chaotic and the other
one is nonchaotic).
[1]
W. Lim and S.Y. Kim, "Strange nonchaotic bursting in the quasiperiodically forced HindmarshRose neuron", J. Korean Phys. Soc. 57, 1356~1362 (2010).
[2] W. Lim and S.Y. Kim "Strange nonchaotic spiking in the quasiperiodicallyforced
HodgkinHuxley neuron," J. Korean Phys. Soc. 57, 2329 (2010).
¡¡
Strange Nonchaotic
Response in the Quasiperiodicallyforced Neuron
We
study dynamical responses of the selfoscillating neuron under quasiperiodic
stimulation. For the case of periodic stimulation on the selfoscillating
neuron, a transition from a periodic to a chaotic oscillation occurs through
period doublings. We investigate the effect of the quasiperiodic forcing on this
perioddoubling route to chaotic oscillation. In contrast to the
periodicallyforced case, a new type of strange nonchaotic (SN) oscillating
states (that are geometrically strange but have no positive Lyapunov exponents)
is thus found to appear between the regular and chaotic oscillating states.
Strange fractal geometry of these SN oscillating states, which is characterized
in terms of the phase sensitivity exponent and the distribution of local
finitetime Lyapunov exponent, leads to aperiodic ¡°complex¡± spikings. Diverse
routes to SN oscillations are found, as in the quasiperiodically forced logistic
map.
[1]
W. Lim and S.Y. Kim, "Strange
nonchaotic oscillations in the quasiperiodically forced HodgkinHuxley neuron,"
J. Phys. A 42, 265103 (2009). [PDF]
[2] W. Lim, S.Y. Kim, and Y. Kim, "Strange nonchaotic responses of the
quasiperiodically forced MorrisLecar neuron," Prog. Theor. Phys. 121, 671683
(2009). [PDF]
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