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Neuroscience Research 64 (2009) 177–184

Contents lists available at ScienceDirect

Neuroscience Research
journal homepage: www.elsevier.com/locate/neures

Analysis of ongoing dynamics in neural networks
Toru Yanagawa a,b, Ken Mogi a,b,*aSony Computer Science Laboratories, Takanawa Muse Bldg., 3-14-13, Higashigotanda, Shinagawa-ku, Tokyo, 144-0022, Japan
Department of Computational Intelligence and System Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta-cho,
Midori-ku, Yokohama, 226-8503, Japanb

A R T I C L E I N F O

A B S T R A C T

Article history:
Received 28 January 2008
Received in revised form 19 February 2009
Accepted 24 February 2009
Available online 9 March 2009

Spontaneous neural activities in the cerebral cortex exhibit complex spatio-temporal patterns in theabsence of sensory inputs [Arieli, A., Shoham, D., Hildesheim, R., Grinvald, A., 1995. Coherent spatiotemporal patterns of ongoing activity revealed by real-time optical imaging coupled with single-unit
recording in the cat visual cortex. J. Neurophysiol. 73, 2072–2093; Arieli, A., Sterkin, A., Grinvald, A.,
Aertsen, A., 1996. Dynamics of ongoing activity: explanation of the large variability in evokedcortical responses. Science 273, 1868–1871], wandering among the intrinsic set of cortical states[Tsodyks, M., Kenet, T., Grinvald, A., Arieli, A., 1999. Linking spontaneous activity of single corticalneurons and the underlying functional architecture. Science 286, 1943–1946; Kenet, T., Bibitchkov,
D., Tsodyks, M., Grinvald, A., Arieli, A., 2003. Spontaneously emerging cortical representations ofvisual attributes. Nature 425, 954–956]. Elucidating the nature of such spontaneous activities is oneof the most intriguing challenges in the effort to understand the computational principles employedby the brain. The precise mechanism behind these salient phenomena, however, is not known. Here
we model the ongoing dynamics of generic neural networks with attractor states using a
conductance-based neuron model. Our realistic modeling shows the existence of up-states anddown-states in the membrane potential, where the up-states exist as spatially clustered patches
moving within the network. Our analysis shows that up-states are sustained by the balance between
excitatory and inhibitory inputs. Synaptic depression and depolarization-dependent potassiumchannels can cause the transitions from the up-states to down-states by affecting the dynamics indifferential manners. The velocity of patches depends on the ﬁring frequency of excitatory neurons
affected by contributing factors. These results suggest that the switching dynamics can be producedby the interactions within the local network, revealing the constraints on the nature of autonomous
dynamics within the cortex.
ß 2009 Published by Elsevier Ireland Ltd and the Japan Neuroscience Society.

Keywords:
Hodgkin–Huxley equation
Excitatory–inhibitory network
Mexican-hat interaction
Ongoing activity
Synaptic depression
Depolarization-dependent potassiumchannels

1. Introduction
The cortical circuit is activated spontaneously even when there
is no or little sensory input. From the empirical point of view,
spontaneous neural activities (Arieli et al., 1995, 1996; Tsodyks
et al., 1999; Kenet et al., 2003) and behaviors (Maye et al., 2007)
represent the most salient and interesting differences between the
biological brain and the artiﬁcial computer, supporting ﬂexible anddynamic perception of the environment. On the larger corticalscale, the existence of the ‘‘default’’ system has been suggested toplay an important role in the functionality of the brain, in which* Corresponding author at: Sony Computer Science Laboratories, Fundamental
Research Lab, Takanawa Muse Bldg., 3-14-13, Higashigotanda, Shinagawa-ku,
Tokyo, 144-0022, Japan. Tel.: +81 3 5448 4380; fax: +81 3 5448 4273.
E-mail address: kenmogi@qualia-manifesto.com (K. Mogi).

baseline activities are sustained by speciﬁc cortical areas (Raichle
et al., 2001; Vincent et al., 2007). At the scale of local networks, the‘‘ongoing activities’’ exhibit complex spatio-temporal patterns thatswitch between a set of intrinsic cortical states (Tsodyks et al.,1999), reﬂecting the overall cortical architecture. Kenet et al.
(2003) observed the ongoing activities in the visual cortex ofanesthetized cats using voltage-dependent optical imaging andfound that the ongoing activities went through cortical states
coincident with orientation columns.
There are two possible basic mechanisms underlying theongoing activities and their transitions (Goldberg et al., 2004). The
ﬁrst assumes that the ongoing activities are driven by cortical noise
and are merely the manifestations of the dynamics of a single
external background state (single-state hypothesis). The second
assumes that the ongoing activities reﬂect the speciﬁcs of internal
cortical interaction, with the cortical state wandering through aseries of intrinsic states as a result (attractor-state hypothesis).

0168-0102/$ – see front matter ß 2009 Published by Elsevier Ireland Ltd and the Japan Neuroscience Society.
doi:10.1016/j.neures.2009.02.011

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Theoretical studies show that there is a trade-off between
mechanisms that support the single-state hypothesis and thosethat support the attractor-state hypothesis. In the case of thesingle-state hypothesis, the ongoing activities are driven bycortical noise external to the network in question. The membranepotentials of neurons in the simulation rise transiently by thenoise, and fall according to the decay time constant of themembrane potential, characterized by a fast transition (<100 ms).
The correlation of membrane potential over space tends to besmall, since the interactions within the local cortical network are
weak. Cai et al. (2005) constructed a large-scale neural network of
the visual cortex. They controlled the coupling strength of longrange connection and reproduced the ongoing activities withtransient strong spatial correlation. They suggested that theongoing activities in visual cortex could be accounted for bymechanisms other than those based on the attractor-statehypothesis. Speciﬁcally, they introduced the scenario of anintermittent desuppressed state (IDS), which is a dynamic stateof high conductance, strong inhibition, and large ﬂuctuations thatarise from intermittent spiking events that are strongly correlated
in time as well as in orientation domains, with the correlation timeof the ﬂuctuations controlled by the NMDA decay time scale. In thecase of attractor hypothesis, on the other hand, the ongoing
activities are sustained by recurrent inputs within the network,where the spatial correlation of membrane potential tends to belarge. In a system characterized by the attractor hypothesis, thetime scale of transition tends to be long (>100 ms), due to the timerequired to break the stability of the up-state.
Spontaneous neural activities involving the up-states and thedown-states are ubiquitously found in the biological brain. During
the naturally occurring slow wave sleep (SWS) in non-anesthetizedcats, there are oscillations (<1 Hz) between the depolarized andhyperpolarized phases. In the up-states, neural activities are
sustained by strong recurrent excitation balanced by the inhibition
involving some intrinsic channels. In the down-states, a majority ofcortical neurons are hyperpolarized, possibly caused by thesynaptic depression and the depolarization-dependent potassiumchannels (Compte et al., 2003; Bazhenov et al., 2002; Hill andTononi, 2005). Depolarization-dependent potassium channels
have been shown to be essential in reproducing the down-states(Compte et al., 2003; Hill and Tononi, 2005).
Incorporating the up-states and down-states as signiﬁcantingredients is compatible with the attractor-state hypothesis inwhich the up-states could be sustained by recurrent corticalinteractions. Such a model needs to reproduce two properties of
spontaneous activities. The ﬁrst is that the neural activities havepatch-like spatial correlation, reﬂecting the cortical architecture.
The second is that the neural activities exhibit transient changes,
characterized by time scales corresponding to the physiological
data.
Here we hypothesized that the switching between the corticalstates of spontaneous ongoing activities in visual cortex is theresult of the transition dynamics between the up-states and downstates. We investigate the transitions within the ongoing dynamics
in a generic network by combining the biophysical neuron modelwith the Mexican-hat type interaction (i.e., excitation dominatingon the short spatial range followed by inhibition on the longer
range), incorporating synaptic depression and depolarizationdependent potassium channels. In the local circuitry of primaryvisual cortex (V1), the Mexican-hat type interaction causes
recurrent interaction leading to cortical states selective toorientations (Ben-Yishai et al., 1995; Ernst et al., 2001). The
synaptic depression and the depolarization-dependent potassiumchannel cause the transitions from the up-states to the downstates.Our simulation shows that the transition velocity of ongoing
activities in the visual cortex can be reproduced by adjusting theparameters of synaptic depression and depolarization-dependent
potassium channels, despite the general tendency of slow
transitions in neural network models based on the attractor-statehypothesis. The necessary ﬁne-tuning of parameters involved puts
considerable constraints on the makeup of the network. We
discuss the differential effects of synaptic depression anddepolarization-dependent potassium channels on the ongoing
activities. The velocities of traveling waves are rate-limited by thetransition from the up-state to the down-state, contrary toprevious models (Compte et al., 2003; Bazhenov et al., 2002;
Golomb and Ermentrout, 2001, 2002) in which the transition fromthe down-state to the up-state is rate-limiting. Finally, we suggest
to test the relevance of mechanisms based on the attractor-statehypothesis for biological networks such as the primary visualcortex by adjusting the parameters of synaptic depression anddepolarization-dependent potassium channels.
2. Methods
We used Hodgkin–Huxley equations similar to those describedby Compte et al. (2003). The pyramidal neurons were modeled asconsisting of two compartments, the soma and the dendrite. The
spiking currents INa and IK were located in the soma, together witha leak current IL, a fast A-type K+ current IA, a non-inactivating slow
K+ current IKS and a depolarization-dependent potassium current
IKNa. The dendrite contained a high threshold Ca2+ current ICa, a
Ca2+-dependent K+ current IKCa, a non-inactivating (persistent) Na+
current INaP, an inward rectiﬁer (activated by hyperpolarization)
non-inactivating K+ current IAR. The depolarization-dependent
potassium channel appears to cause the termination of thedepolarized phase of the slow oscillation (Sanchez-Vives andMcCormick, 2000). This channel is activated by the inﬂux ofsodium ion that accumulate during the period of depolarizationand spiking.
Interneurons consisted of a single compartment, with spiking
currents INa and IK, and a leak current IL. These parameters were
given explicit treatment within the dynamics described by theequations.
Excitatory synaptic current consisted of an AMPA-mediatedcurrent and an NMDA-mediated current (Jahr and Stevens, 1990;
Wang, 1999). Inhibitory synaptic current was modeled as a GABAmediated current (Wang, 1999). A short-term synaptic depression
was introduced for pyramidal-to-pyramidal recurrent excitatory
connections (Tsodyks and Markram, 1997; Abbott et al., 1997). The
synaptic depression reduces the synaptic current in response to thecontiguous spikes of presynaptic neurons and recovers it in theabsence of presynaptic activities.
The initial values for membrane potentials were randomly
distributed around the mean values of 50 mV and 60 mV for theexcitatory and inhibitory neurons, respectively, with a range of10 mV. No delay was explicitly assumed in the transmission
between the neurons.
For further details about the models used in the simulation,refer to the supplementary material (Text S1).
The generic network consisted of 2304 pyramidal neurons and576 interneurons. The pyramidal neurons were arranged on aregular square grid (Fig. 1(c)). Inhibitory neurons were positioned
on every second grid points, half a grid out of alignment fromexcitatory neurons. The network had a periodic boundary
condition, leading to a 2D torus topology. The excitatory andinhibitory synaptic connections were given by a Gaussian
distribution. Following previous works (Ben-Yishai et al., 1995;
Ernst et al., 2001; Crook et al., 1998), we assumed that theexcitatory and inhibitory synaptic connections induce lateralinteractions having a Mexican-hat type shape. Standard deviationof the Gaussian was set to be 2 grids and 4 grids for the excitatory

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179

Fig. 1. (a) Distribution of presynaptic excitatory neurons as ‘‘seen’’ from an excitatory neuron. Gray scales represent the average distribution of excitatory neurons. The scale
bar represents the probability values between 0 and 1.0. Open circles represent the typical actual distribution of excitatory neurons. (b) Presynaptic inhibitory neurons ‘‘seen’’
from an excitatory neuron. Gray scales represent the average distribution of inhibitory neurons. The scale bar represents the probability values between 0 and 0.2. Opencircles represent the typical actual distribution of inhibitory neurons. (c) The network architecture. Filled circles represent the excitatory neurons. The open circles represent
the inhibitory neurons. (d) A snapshot of the membrane potentials of the soma in the excitatory neurons. (e) A snapshot of the membrane potentials of the soma in the
excitatory neurons after spatial interpolation. The black lines represent the borders between up-states and down-states. The centroids of the patches are represented by thesymbols ‘‘’’.

(sex) and inhibitory (sinh) synapses, respectively. The number ofpresynaptic neurons was set to be constant. For pyramidal andinhibitory neurons, the number of presynaptic excitatory neurons
was 28, while that for presynaptic inhibitory neurons was 4. As aresult of the assignment process following the Gaussian distribution constrained by the ﬁxed number of presynaptic neurons, thenumber of postsynaptic neurons had some degrees of variance inthe process of random selection for the target neurons. For the
pyramidal neuron, the number of post-synaptic pyramidal neurons
was 28  3 while that for inhibitory neurons was 7  1.7. For the
inhibitory neuron, the number of post-synaptic pyramidal neuron
was 16  2.9 and that of inhibitory neurons was 4  1.9.
The model was implemented in a C++ code and simulated using
a forth-order Runge-Kutta method with a time step of 0.06 ms.
3. Results
The synaptic depression and the depolarization-dependent
potassium channels are two possible candidates for the mechanism that causes the transition from the up-state to the down-state.
The activation of depolarization-dependent potassium channel
results in the hyperpolarization of excitatory neurons (Compte
et al., 2003; Hill and Tononi, 2005). Therefore we tested two
conditions. Firstly, we simulated the network with the synaptic

depression only (condition1). Secondly, the depolarization-dependent potassium channel was introduced to the network in addition
to the synaptic depression (condition 2).
First we describe the simulation based on condition 1. Since thereversal potential and peak conductance of the leakage current is
distributed randomly (see Text S1), some pyramidal neurons are
spontaneously active with low ﬁring rates (0.2  0.4 Hz). This
arrangement is based on the observation of spontaneously ﬁring
activities even after blocking the glutamatergic excitatory postsynaptic potentials in in vitro experiments (Sanchez-Vives andMcCormick, 2000). This is the only source of spontaneous activity
in this simulation.
In condition 1, when a random initial condition was given,
spontaneously active neurons ﬁred together occasionally, which
triggered a cascade of recurrent excitation that locally brought the
network into the ﬁring regime of the up-states (see Text S5). The
local Mexican-hat type interaction between excitatory andinhibitory neurons generated a spatially heterogeneous (patchy)
distribution (Fig. 2(g)). After the network went into the steadystate, the patches of up-states started to move (Video S16, http://
www.qualia-manifesto.com/Video_S16.mov). The membranepotential showed a robust oscillation (see Text S10). The
distribution of the histogram of the membrane potential of
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Fig. 2. (a) The membrane potential of a soma in the pyramidal neuron before (black
trace) and after (blue trace) spatial interpolation. The threshold between the upstate and down-state is shown in the red trace. (b) A typical time course of thetransition between the up-state and down-state of a neuron. (c) A typical timecourse of the synaptic current of GABA (blue trace), AMPA (red trace), and NMDA
(black trace). (d) A typical time course of all intrinsic currents except for INa and IK in
excitatory neurons. (e) A typical count of excitatory and inhibitory synaptic spikes.
(For interpretation of the references to color in this ﬁgure legend, the reader isreferred to the web version of the article.)

invariant (Fig. S12). An approximately constant ratio of the
population is found to be in the up-state (deﬁnition of up-stateand down-state given below). In the steady state, the inhomogeneity introduced by random distribution of leakage currents isaveraged by the network interaction, while the ﬂuctuations ofpatch dynamics are affected by cortical interaction.
To extract and visualize the patches of activities, membranepotential of each neuron was averaged with the neighboring eightneurons (Fig. 1(e)). The Mexican-hat type interaction generated

regions of positive and negative spatial correlation (Fig. 3(b)). If weassume that the ratio (in terms of distance) of positive and negative
regions is 1:1 as suggested by the data, the ratio (in terms of areas)of the positive and negative regions, each represented byconcentric circles, is 1:3. As an estimate based on this result, wedeﬁned the ﬁrst upper quartile of population in the distribution
over the membrane potential value range as belonging to the upstate, while the remaining population was judged to be in the
down-state (Fig. 2(a) and (b)). This distinction was then used forthe visualization of the patches and the calculation of moving
velocities for the centroids of the patches. This distinction isinvariant over time (Fig. S13).
In our simulation, there are no clearly distinctive peaks in themembrane potential distributions of individual neurons (Fig. S14).
Since there is no a priori reason why a particular threshold value
should be adapted for the border between the up-state and thedown-state, the deﬁnition above is in essence arbitrary. Notwithstanding this reservation, the dynamics of the network is found to
be different for the upper and lower regions of the membranepotential, with the threshold deﬁned above as the effective border.
The activities of the synaptic currents and intrinsic channel
currents have different modes for the up-states and down-states(Fig. 2(c) and (d)). In the up-state, the AMPA-mediated, NMDAmediated, and GABA-mediated currents are activated (Fig. 2(c)).
The intrinsic channel currents, the non-inactivating slow K+
current IKS, and a non-inactivating (persistent) Na+ current INaP,
both of which are voltage-dependent channels, also get activated.
In addition, the Ca2+-dependent K+ current IKCa becomes open.
In the down-state, the number of excitatory synaptic inputs are
smaller than that for the up-state (Fig. 2(e)). However, theinhibitory synaptic inputs are as active as in the up-state, which is
caused by the Mexican-hat type interaction (Fig. 2(e)). The inward
rectiﬁer non-inactivating K+ current IAR (which is activated byhyperpolarization) is also activated.
Synaptic depression is induced in the excitatory synaptic
connections between excitatory neurons. In the up-state, theexcitatory synaptic current gradually decreases as the neurons
keep ﬁring, which in turn causes a decrease in the ﬁring frequency.
There exits a point where the balance between the recurrent
decrease of excitatory synaptic input and the constant inhibitory
synaptic input is broken (Fig. 3(a)–(i)). The term of voltagedependence in the NMDA-mediated current, which is drasticallyreduced as the membrane potentials decrease, also contributes tothe switching from the up-state to the down-state.
To investigate the robustness of the observed behavior of thesystem, we modiﬁed the following parameters: a ¼ ðg 0NMDA EE
EE
EE
EE
EI
EI
=gAMPAÞ=ðgNMDA=gAMPAÞ ¼ ðg 0NMDA EI=g 0AMPA EIÞ=ðgNMDA=gAMPAÞ, whichchanges the strength of NMDA-mediated current, and b ¼
EI
g 0AMPA EI=gAMPA
which affects the strength of inhibitory feedback.
EE
In all conditions, gAMPA
was kept constant. Namely, g 0NMDA EE ¼
EE
EI
EI
a gNMDA, g 0AMPA EI ¼ b gAMPA, and g 0NMDA EI ¼ a b gNMDA.
As a measure of the heterogeneity between the up-states andthe down-states, the spatial correlation coefﬁcients between themembrane potentials of excitatory neurons were calculated. The
spatial correlation for the patches of up-state is found to depend on
the strength of NMDA-mediated current (Fig. 3(b) and (c)).
The parameter b controlled the strength of the inhibitory
feedback. When b was large, the strength of inhibitory feedback
became correspondingly large, and the spatial correlations
decreased (Fig. 3(c)). On the other hand, when the inhibitory
feedback became smaller, the network came close to an overexcited ‘‘epilepsy’’ state, with the spatial correlation becoming
smaller.
The moving velocity was calculated for the trajectory of thecentroid of the patches in the network (Fig. 1(e)). Here, a patch was
deﬁned as the cluster of connected up-states. The average moving

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181

Fig. 3. (a) A typical time course of the average membrane potential of the soma of excitatory neurons and the average count of synaptic excitatory and inhibitory spikes when
the neuron goes under the transition from the up-state to the down-state. Time bin is 1.5 ms. (a-1) represents condition1 (synaptic depression only). (a-2), (a-3), and (a-4)
represent condition 2 (synaptic depression and depolarization-dependent potassium channels), with the following parameters (a-1) a = 1, b = 1, Pv = 0.3 (a-2) a = 1, b = 1.33,
Pv = 0 (a-3) a = 1, b = 0.8, Pv = 0.3 (a-4) a = 1, b = 1.33, Pv = 0.5 and IKNa = 0. These parameter values are selected so that the ﬁring rates of excitatory neurons become 10 Hz. (b)The average correlation coefﬁcient of the membrane potential between neurons as a function of distance. Parameter a = 0.5, 1, and 2 (b = 1). (c) Correlation coefﬁcient at the
distance of 1 grid for some values of parameters a and b. (d) The distance traveled as a function of the lifetime of patches with a = 1 and b = 1. The straight line depicts the resultof linear regression. (e) The average ﬁring rate of excitatory neurons and the average moving velocities for some values of parameters a and b (Pv = 0.3). (f) The average ﬁring
rate of excitatory neurons and the average moving velocities for some values of a and Pv (b = 1). (g) The average ﬁring rate of excitatory neurons and the average moving
velocities for some values of a and Pv (a = 1) in condition 2 (synaptic depression and depolarization-dependent potassium channels). (h) The moving velocity as a function of
the average ﬁring rate of excitatory neurons for some values of parameters a and b with Pv = 0.1, 0.3, and 0.5 in condition 1 (1) and condition 2 ((2) and (3)). IKNa = 0 in (3).

velocity of the patches was found to depend on the ﬁring
frequencies and the modes of synaptic depression. The moving
patches repeatedly went through the process of generation anddisappearance, while the ratio of the neural population in the upstate was roughly preserved (details described below and in the
supplementary online material).
Here’s a qualitative description of what is typically observed in
the simulation. A patch moves continuously, being affected by andinﬂuencing in turn the motion of the surrounding patches. During
its existence, the moving velocity has a certain degree of variation.
When a patch is ‘‘stuck’’ with the surrounding patches, it slows
down. Sometimes such a ‘‘trafﬁc jam’’ can cause the disappearanceof a patch. When a space is created in the vicinity of a patch, it‘‘moves’’ for that space with an accelerated speed. There is thetendency that after the disappearance of a patch, the nearest patch
splits into two patches, one of which moves for the space madevacant by the disappearance. At present, we are not aware of any

physiological data that correspond to these observed phenomenain the simulation.
Fig. 3(d) shows the relation between the lifetime of each patch
and the distance traveled. A series of centroids at different times
are regarded as belonging to one patch trajectory within the rangeof its spatio-temporal continuity. The split of a patch into two
initiates a new series of trajectory. The average velocity is
calculated for a single trajectory of each patch thus deﬁned.
In our simulation, the values of parameters sex and sinh affectthe size of patches. Changing the values of sex, sinh and the network
size while keeping other parameters ﬁxed resulted in changes ofthe patch sizes and their transition velocities. The transition
velocity of the patches is an important parameter to be comparedwith the physiological data. The velocity of a patch increases
linearly, following an increase of the patch size (see Supplementary
data, Text S2). The dynamics on which we focused in this study do
not acutely depend on the number of neurons contained in theAuthor's personal copy
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patch, but rather on its diameter deﬁned by the length of sex andsinh (Fig. S4). The patch diameter, 12 grids in our simulation,roughly corresponds to the 1 mm size of a column in the biologicalnetwork (Tsodyks et al., 1999). The total size of the model network
thus roughly corresponds to 4 mm  4 mm.
An increase in parameter a and a decrease in parameter b led tothe increase in both the ﬁring rate and the moving velocity(Fig. 3(e)). When the synaptic depression parameter Pv, (the rate ofdecrease in the current after a spike) was made larger while theﬁring rate was kept small, the velocity of the patch became larger(Fig. 3(f)). Fig. 3(h) shows the relation between the ﬁring rate andthe transition velocity for the patches in the parameter space. The
transition velocity is observed to become faster as the ﬁring rate
and the synaptic depression parameter Pv become larger.
To the best of our knowledge, there are no physiological data
explicitly addressing the traveling velocity of ongoing activities
resulting from switching cortical states. Based on the available
data, we estimate the traveling velocity as follows. The decay
constant of autocorrelation function calculated from the timeseries of correlation coefﬁcients between the evoked map andsingle frames from spontaneous activities is 80 ms (Kenet et al.,2003). When approximated linearly, the time in which thecorrelation becomes zero is 127 ms (80 ms divided by 0.63). Thisestimated time would be shorter than the actual time since theautocorrelation function is convex downwards. If we assume thatthe patch sweeps an area corresponding to a column of 1 mm indiameter, and that the cortical states would have undergone thetransition from the up-state to the down-state or vice versa in thattime, the traveling velocity would be 7.9 mm/s. We use this value
as a rough estimate of the velocity of traveling waves in vivo.
In an effort to match the physiological value for the travelingvelocity of the patches estimated above, we conducted further
simulations with condition 2, where depolarization-dependent
potassium channels were introduced in addition to synaptic
depression. Along with the addition of depolarization-dependent
potassium channels to the network, which enhances the inhibition
effect in proportion to ﬁring rate during the up-state, someparameters were changed from condition 1. Firstly, the permeability of the leak current IL was reduced by decreasing the
conductance and increasing the reversal potential of the leakcurrent. This change effectively increases the inﬂux of thepotassium current during the down-state and functions to shorten
the duration of the down-state, preventing the network from being
in the down-state overall by the effect of IKNa. Physiologically, thisarrangement corresponds to the blocking of leakage currents
arising from the increase of neuromodulators such as acetylcholine
in the transition from the sleeping state to awake state(McCormick, 1992). Secondly, the inward rectiﬁer non-inactivating
K+ current IAR was set to be zero, as it lengthens the duration of thehyperpolarization in the down-state. The effect of IAR would
become otherwise stronger in the existence of IKNa, forcing all
neurons to the down-state. Thirdly, the conductance of noninactivating (persistent) Na+ current INaP was made larger, making
it easier for IKNa to break the balance between the excitatory andinhibitory currents.
Typical dynamics of the ongoing activity in condition 2 is shownin Fig. S18 and Movie S17 (http://www.qualia-manifesto.com/
Video_S17.mov). The membrane potentials become variable incondition 2, due to the effect of depolarization-dependent
potassium channels causing a strong hyperpolarization (Fig. S13(b)). This situation makes the deﬁnition of up-state and down-state
adopted for condition 1 impractical, making the border unstable inits designated properties. In condition 2 therefore, the border
between the up-state and down-state was set at a constant value ofmembrane potential, 65 mV (Fig. S14 (b)), which was found to beeffective in extracting the centroids of patches.

The decay of membrane potential in the transition from the upstate to down-state became faster when the depolarizationdependent potassium channels were incorporated (condition 2)
compared to the situation with synaptic depression only (condition 1) (Fig. 3a). These results suggest that depolarizationdependent potassium channels are more effective than synaptic
depression in terminating the up-states. In the presence of
depolarization-dependent potassium channels, the ﬁring rate
and the velocities of the patches became smaller as the synaptic
depression parameter Pv, became larger (Fig. 3(g)).
Finally, Fig. 3(h) shows that the velocities of patches are larger
in the presence of synaptic depression and depolarizationdependent potassium channels than those in the presence ofsynaptic depression only. The ﬁring rate was modiﬁed bymanipulating the parameter b, the amplitude of inhibitory
feedback to excitatory neurons. The patch velocity in the presenceof depolarization-dependent potassium channels (condition 2) isclosest to the physiological data (with a maximum velocity of8.0 mm/s). As parameters IL, IAR and INaP were modiﬁed alongwith the introduction of depolarization-dependent potassiumchannels in condition 2, the effect of these parameter changes
alone was checked by setting the value of IKNa to zero. As can beseen from Fig. 3(h), these changes in parameters by themselves
cannot make the velocities comparable to the physiological level.
4. Discussion
In this study, we constructed a network of spiking neurons with
up-states and down-states in the presence of synaptic depression
and depolarization-dependent potassium channels. We havedescribed some robust properties of the ongoing activities
characterized by transitions between cortical states. Our study
focused on the nature of the mechanism involved in the ongoing
activities exhibited in, e.g., the primary visual cortex (Arieli et al.,
1995, 1996; Tsodyks et al., 1999; Kenet et al., 2003).
The results reported here describe the properties of a genericneural network. The architecture in our model has an isotropic
local connectivity which follows a Gaussian distribution. Several
evidences suggest that such a rudimentary architecture canactually be regarded as a founding basis for the biological neuralnetwork. Although functional maps are organized at a ﬁne scale in
the primary visual cortex (V1) (Tsodyks et al., 1999), it is known
that cortical neurons have isotropic local connectivity even at thepinwheel centers (Marino et al., 2005). A network model of V1 withisotropic connectivity have been shown to exhibit orientation
selectivity (Ernst et al., 2001). Based on these evidences, wesuggest that our present model reproduces some of the basic
properties of functionally organized biological neural networks
such as V1. The dynamics of speciﬁc neural network architecture in
the brain is to be studied on top of these basic ﬁndings.
Patchy activities are observed in physiological conditions(Tsodyks et al., 1999; Kenet et al., 2003). The transition of thepatches to neighboring spaces observed in the simulation isconsistent with the physiological data which suggest that theongoing activities have a tendency to make transitions to thecortical areas representing neighboring angles (Kenet et al., 2003).
At present, the exact relation between the traveling wave
measured in the cortical slices (Petersen et al., 2003; SanchezVives and McCormick, 2000) and the switching of ongoing
activities in V1 (Tsodyks et al., 1999; Kenet et al., 2003; Goldberg
et al., 2004) is not known. In the model studies, if the network has
no Mexican-hat type interaction, the transition velocity is known
to depend on the transition from the down-state to the up-state(Compte et al., 2003; Bazhenov et al., 2002; Golomb and
Ermentrout, 2001, 2002). We have shown here in a generic neuralnetwork model that when the network has a Mexican-hat type

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T. Yanagawa, K. Mogi / Neuroscience Research 64 (2009) 177–184

interaction, the process of transition from the up-state to thedown-state sets a limit to the transition velocity, as the patch
cannot start to move if there is no synaptic depression (see Fig. 3(f),except for Pv = 0). There is no signiﬁcant effect of noise fromspontaneously ﬁring neurons on the transition. The rate-limiting
signiﬁcance of the transition from the up-state to the down-staterather than that from the down-state to the up-state indetermining the velocity of traveling waves is one of the novel
aspects in the present model. In models based on the attractor
hypothesis studied previously (e.g., Cai et al., 2005), the introduction of synaptic depression has not been shown to result in asigniﬁcant effect. Hill and Tononi (2005) did not explicitly quantify
the effect of synaptic depression on the traveling wave velocity ofongoing spontaneous activities. In our model, the decrease insynaptic current by the synaptic depression and the resulting
breakdown of the up-states is instrumental in realizing the moving
patches of activities, as bistability has been introduced in themembrane potential through intrinsic ion channels.
As already mentioned, the mechanism based on the ‘‘singlestate’’ hypothesis has been suggested to account for the coherent
ongoing activities observed in the primary visual cortex (Goldberg
et al., 2004). In a model adapting this hypothesis, spontaneousongoing activities are the results of the dynamics of a single
background state, driven by the cortical noise. In a model based on
the attractor-state hypothesis, on the other hand, the transition
velocity of the patch is constrained by the process of termination of
the up-state.
The synaptic depression and depolarization-dependent potassium channels are possible candidates to trigger the transitions
from the up-states to the down-states, affecting the ongoing
activities in differential manners. Parameter sets (2) and (3) inFig. 3(h) suggest that the effect on the traveling velocity is greaterin the case of depolarization-dependent potassium channels thansynaptic depression. In addition, the introduction of depolarization-dependent potassium channels is found to interfere with theeffect of synaptic depression. In the case of synaptic depression
only (Fig. 3(f)), larger rates of synaptic depression lead to larger
traveling velocities. In contrast, when the synaptic depression anddepolarization-dependent potassium channels coexist, larger rates
of synaptic depression lead to smaller values of traveling velocities(Fig. 3(g)). The reason is as follows. The potassium current IKNa
increases in proportion to the accumulated number of spikes of
excitatory neurons. The synaptic depression effectively decreases
the amplitude of presynaptic excitatory current in proportion to
the accumulated number of spikes of excitatory presynaptic
neurons. The introduction of synaptic depression makes themembrane potential and the number of spikes in the up-state
lower because of a decreased inﬂux of current, compared to thecondition of no synaptic depression (Fig. 3(a-2) and (a-3)). A lower
number of spikes per unit time leads to a lower rate of IKNa, making
it take longer to break the balance between excitatory andinhibitory currents in the membrane potential. As a result, thesynaptic depression diminishes the effect of depolarizationdependent potassium channel, leading to lower traveling velocities.
There are some ways to test the validity of the current model.
Because of the nature of transitions between the up-states anddown-states in the attractor-state model described here, affecting
the mechanism of termination of the up-states would lead tocorresponding changes in transition velocities. This point could bechecked by changing the parameter involved in synaptic depression. The application of 4-aminopyridine (4-AP) or a highconcentration of Ca2+ is known to accelerate the short-term
synaptic depression (Varela et al., 1997). Our model predicts thatwhen the synaptic depression parameter Pv is decreased, thetransition velocity increases in the coexistence of synaptic

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depression and depolarization-dependent potassium channels(Fig. 3(g)), and decreases in the condition of synaptic depression
only (Fig. 3(f)). If there is no change in the transition velocity
corresponding to the modiﬁcation of synaptic depression, there are
two possible interpretations. The ﬁrst is that the depolarizationdependent potassium channels rather than the synaptic depression are responsible for causing the transition (condition (2) inFig. 3(h), with Pv = 0), in which case the attractor-state hypothesis
would be maintained. The second is that mechanisms other thanthose based on the attractor-state hypothesis, e.g., the single-statehypothesis, or the intermittent desuppressed state (Cai et al., 2005)
are relevant to the physiological condition, with the introduction ofsynaptic depression not affecting the transition velocities.
In conclusion, the results reported here shed a new light into thephysiological basis of spontaneous neural activities in the cortex.
The ongoing activities sustained by the internal dynamics leads toa larger degree of freedom for the conﬁguration of the cortical
network involved in various computations. The intrinsic ability of
the local network to undergo transitions between multiple stateson its own might have an important role in salient cognitive
functions, e.g., the memory consolidation process during sleep(Huber et al., 2004; Stickgold et al., 2000). The results of oursimulation, taken as a whole, suggest that parameters in the
dynamics of the generic network needs to be ﬁne-tuned in order toreproduce the observed behavior of the biological neural network.
The circuits in the brain therefore is likely to be operating in anarrow domain in the relevant parameter space, giving important
constraints on cortical dynamics.
Acknowledgments
We thank (in alphabetical order) Toru Aonishi, Koji Ito,
Yoshihiro Miyake, Eizo Miyashita, Kiyohiko Nakamura, Fumihiko
Taya, and Masayuki Yamamura for helpful discussions. We are
grateful for two anonymous reviewers for their insightfulcomments on an earlier version of our paper.

Appendix A. Supplementary data
Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.neures.2009.02.011.
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