Download Sustained activity with low firing rate in a recurrent network

Neurocomputing 44–46 (2002) 473 – 478
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Sustained activity with low ring rate in a
recurrent network regulated by
spike-timing-dependent plasticity
Katsunori Kitanoa; ∗ , Hideyuki Câteaub , Tomoki Fukaia; b
a Department
of Information-Communication Engineering, Tamagawa University,
6-1-1 Tamagawagakuen, Machida, Tokyo 194-8610, Japan
b CREST, JST (Japan Science and Technology Corporation), 6-1-1 Tamagawagakuen, Machida,
Tokyo 194-8610, Japan
Abstract
In order to study roles of spike-timing-dependent plasticity (STDP) at the network level,
we applied STDP to a model of the cortical recurrent network. We found that STDP brought
self-organization of a bistable neural activity that is essential for the working memory function.
Furthermore, our simulations showed that the typical two ring patterns during the persistent
activity were achieved, which depend on the time window of STDP; the short time window
( ∼10 ms) yielded an asynchronous ring activity, whereas the longer one ( ¿20 ms) produced a
c 2002 Elsevier Science
synchronous spike packet propagating along a chain of cell assemblies. B.V. All rights reserved.
Keywords: Long-term potentiation=depression; Sustained activity; Working memory; Synre chain;
Computational model
In cortical and hippocampal pyramidal neurons, spike timing has important information for the synaptic plasticity [2,11]. If an excitatory postsynaptic potential (EPSP)
precedes a postsynaptic action potential, the synapse exhibits long-term potentiation
(LTP), whereas the same synapse undergoes long-term depression (LTD) if the timing is reversal. While the timing dependence of LTP=LTD infers their functional role
in temporal coding, recent studies rather showed their implication in rate coding. Firing rate of a postsynaptic neuron is only moderately changed with presynaptic ring
rate, if synapses are reorganized by the timing-dependent LTP=LTD [12,13,15]. Such
∗
Corresponding author.
E-mail address: [email protected] (K. Kitano).
c 2002 Elsevier Science B.V. All rights reserved.
0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 2 ) 0 0 4 0 4 - 6
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K. Kitano et al. / Neurocomputing 44–46 (2002) 473 – 478
an activity regulation seems to be more useful in networks with recurrent excitation
than in single neurons or networks with feedforward connections. The fact also implies that spike-timing-dependent plasticity (STDP) is responsible for the organization
of networks exhibiting a sustained activity that has been observed in the delay period
of working memory task [6 –8]. It has been known that long lasting currents such as
the persistent Na+ current contribute to the maintenance of the activity with the ring
rate observed experimentally, 20 –50 Hz [10,5]. However, the relation between such an
activity and synaptic connectivity is poorly understood. Therefore, in order to clarify
this point, we constructed the model of cortical network and introduce the STDP rule
formulated by Song et al. into the excitatory recurrent connections [13].
Our network model consists of NE = 200 excitatory and NI = 50 inhibitory
neurons. The dynamics of each neuron is described
with the Hodgkin–Huxley
equation: Cm dV=dt = −gL (V − EL ) − INa − IK − j Isyn; j + Iapp + Inoise with Cm =
3:0 F=cm2 (excitatory) or 1:2 F=cm2 (inhibitory), gL =0:14 mS=cm2 , and EL =−70 mV.
The membrane time constant of the excitatory neuron and the inhibitory neuron turn
to be 21 ms and 9 ms, respectively. The kinetics of the spike generating sodium and
potassium channels follows those of the model by Traub et al. with gNa = 100 mS=cm2 ,
gK = 40 mS=cm2 , ENa = 45 mV, and EK = −80 mV [14]. The neurons are driven by the
synaptic currents that arise from AMPA synapses of the excitatory neurons and from
GABA synapses of the inhibitory neurons. The synaptic currents Isyn is determined by
the rst-order kinetics of the gating variable. We set the activation rate =19:8, the inactivation rate =0:2 (both AMPA and GABA), EAMPA =0 mV, and EGABA =−70 mV
[3]. In the present paper, we assume that only the excitatory-to-excitatory synapses
show the synaptic plasticity. The conductances of this kind of synapses are initially set
at a maximum value gMAX = 0:04 (measured in a unit of the leak conductance) and
transiently change in the range of 0 6 g 6 gMAX through the following rule. If an interval between an EPSP by a presynaptic neuron and an action potential by a postsynaptic
neuron t = tpost − tpre is positive, the conductance of the corresponding synapse is
potentiated as g → g+gMAX Ap exp(−t=p ). Otherwise, g → g−gMAX Ad exp(−|t=d |).
Here, Ap and Ad are the maximum amount of potentiation and depression, respectively.
And p and d represent the time window of potentiation and depression, respectively.
Keeping the area of LTP Ap p = 0:2 and that of LTD Ad d = 0:21, we examined two
cases of the relatively short time window p = d = 10 ms and the relatively long time
window p = d = 20 ms. The other types of the connections are randomly generated.
The connection in a direction between a pair of neurons is adopted with the probability of 0.1, the conductance of which is set at a xed value of 0.04 (AMPA) or 0.05
(GABA). Besides, all the neurons in the network are disturbed by external noise. The
noise is Gaussian white noise with the intensity that induces spontaneous ring activity
with 0.5 –1:5 Hz of isolated neurons.
Our interest is whether the network autonomously organizes the connectivity so as to
produce a self-sustained activity with the STDP rule. Therefore, we rstly let the network run autonomously and then investigated the activity and the synaptic connectivity
of the self-organized network. Fig. 1 shows the time evolution of average ring rate of
the network in the self-organization process. In the early stage of the process, synaptic
modication rapidly proceeded because of strong excitation due to the initial all-to-all
K. Kitano et al. / Neurocomputing 44–46 (2002) 473 – 478
475
average frequency (Hz)
100
80
60
40
20
0
0
20
40
60
80
100
120
time (s)
Fig. 1. Typical examples of the time evolution of ring rate averaged over the network during
self-organization. Note the presence of an initial epoch showing high ring rate and a sudden fall of activity
to a spontaneous discharging state.
Iapp
# of neurons
200
100
0
0
1000
2000
3000
4000
5000
time (ms)
Fig. 2. Raster plot of the sustained activity shows the bistability of the self-organized recurrent network.
The trace above the raster plot represents a depolarizing or hyperpolarizing applied current Iapp to activate
or inactivate the network. Iapp =gL = 70 mV from t = 1000 to 1020 ms, and −70 mV from 4000 to 4020 ms.
Otherwise Iapp =gL = 0 mV.
synaptic conguration. After the stage, slow modication of the synaptic conguration
was accompanied by the gradual decrease in the activity, which continued until the
activity suddenly fell down to spontaneous ring level. The reason for such a transition, which occurred at 20 –30 Hz, is that the recurrent excitation becomes suciently
weak so that the ‘on’-state maintained by the recurrent excitation could be disturbed
by strong hyperpolarizing noise. Once the network entered the spontaneous ‘o ’-state,
no signicant modication of the synapses occurred because of a very low ring rate.
To examine whether the self-organized network works as working memory, we
turned o the STDP rule and transiently stimulated the network. The application of a
transient depolarizing current to 30% of the neurons induced the self-sustained activity
with about 30 Hz. On the other hand, a transient hyperpolarizing current terminated the
‘on’-state activity (Fig. 2). As we can see, the connectivity is found to be responsible
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K. Kitano et al. / Neurocomputing 44–46 (2002) 473 – 478
1
auto correlation
# of neurons
200
100
0.6
0.4
0.2
0
-0.2
0
200
400
600
800
1000
time (ms)
2050
(a)
2100
2150
2200
0.2
cross correlation
0
2000
time (ms)
0.2
fraction in bin
0.8
0.15
0.1
0
-0.1
-0.2
-500
0.1
(b)
0
500
time (ms)
0.05
0
0
(c)
0.2
0.4
0.6
0.8
1
normalized synaptic weights g/gMAX
Fig. 3. The asynchronous sustained activity obtained through the STDP rule with the short time window.
(a) The self-organized network exhibits asynchronous ring in the ‘on’-state. (b) An auto-correlogram for a
neuron and a cross-correlogram for a pair of the neurons show the asynchronous spiking activity. (c) The
distribution of the synaptic weights.
for the bistable activity, i.e., spontaneous ring ‘o’-state and self-sustained ‘on’-state.
Though we demonstrated only the case of a 3 s interval of the ‘on’-state activity, the
longer interval can be produced unless the external noise is intensive.
The self-organization and the sustained activity mentioned above was achieved in
both the case of the short time window for STDP (10 ms) and that of the long time
window (20 ms). However, there was found to be some dierences in the achieved
ring patterns and the self-organized synaptic connectivity. When p = d = 10 ms, Fig.
3a shows that the activity of the network in the ‘on’-state was almost asynchronous,
which was tested by the spike auto-correlogram and cross-correlogram (Fig. 3b). In
addition, we examined how was the achieved distribution of the synaptic weight in this
case. In the single neuron study, it has been reported that STDP brought competition
among the synapses terminating to an identical postsynaptic neuron so that the synaptic
weights were distributed with a bimodal shape [12,13,15]. The activity regulation was
accomplished through such a competitive process. However, Fig. 3c represents the
continuous distribution of the synaptic weights, which means that synaptic competition
occurred only weakly in this case.
On the other hand, when p = d = 20 ms, cell assemblies were self-organized. The
matrix of the achieved synaptic connectivity indicates that three assemblies connected
as a closed loop (Fig. 4a). As shown in Fig. 4b, the synaptic competition strongly
operated, which caused the bimodal distribution of the synaptic weights in this case.
K. Kitano et al. / Neurocomputing 44–46 (2002) 473 – 478
477
200
0.6
150
fraction in bin
index j
0.5
100
1.0
0.8
0.6
50
0.4
0.3
0.2
0.4
0.1
0.2
0
0
(a)
50
100
150
200
0
0
0
(b)
index i
0.2
0.4
0.6
0.8
1
normalized synaptic weight
# of neurons
200
100
0
2000
(c)
2050
2100
2150
2200
time (ms)
Fig. 4. The cell assemblies formed through self-organization by STDP. (a) The weight matrix of the
excitatory-to-excitatory synapses displays a closed loop structure of three cell assemblies. (b) The distribution of the synaptic weights. (c) The raster plot for the spiking activity of the network.
From these results, it is found that causal and acausal relations among the neural
activity were expressed as the feedforward-like structure through the competition. The
raster plot displays that the activity propagated along the closed loop that consisted of
the three cell assemblies (Fig. 4c).
As presented above, it was found that, even at the network level, the STDP rule
has the ability to organize the recurrent connections so as to regulate the activity. As
a result, the two activity patterns, asynchronous spiking activity and the wave of the
synchronous spike packet, were obtained. The bistable activity with the asynchronous
pattern was similar to the delay-period activity observed in the prefrontal cortex during
working memory task. On the other hand, the wave of the synchronous spike packet
can be associated with ‘synre chain’, the feedforward neural network that propagates
a packet of synchronous spikes [1,4]. Though the number of the cell assemblies was
only three in our model, STDP is potentially responsible for the organization of the
synre-chain-like structure [9]. As the reason why the two dierent types of the activity patterns were self-organized, it may be considered that the external noise prevented
STDP from capturing the causal or acausal relations among the spikes. As the time
window is shorter, the noise eect might be enhanced more strongly in comparison.
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K. Kitano et al. / Neurocomputing 44–46 (2002) 473 – 478
Consequently, the stochastic modication due to the noise eect reduced the competition among the synapses, which caused the connectivity for the asynchronous activity.
References
[1] M. Abeles, H. Bergman, E. Margalit, E. Vaadia, Spatiotemporal ring patterns in the frontal cortex of
behaving monkeys, J. Neurophysiol. 70 (1993) 1629–1638.
[2] G.Q. Bi, M.M. Poo, Activity-induced synaptic modications in hippocampal culture: dependence on spike
timing, synaptic strength and cell type, J. Neurosci. 18 (1998) 10,464–10,472.
[3] A. Destexhe, Z.F. Mainen, T.J. Sejnowski, Kinetic models of synaptic transmission, in: C. Koch,
I. Segev (Eds.), Methods in Neural Modeling, MIT Press, Cambridge, 1998, pp. 1–25.
[4] M. Diesmann, M.O. Gewaltig, A. Aertsen, Stable propagation of synchronous spiking in cortical neural
networks, Nature 402 (1999) 529–533.
[5] D. Durstewitz, J.K. Seamans, T.J. Sejnowski, Dopamine-mediated stabilization of delay-period activity
in a network model of prefrontal cortex, J. Neurophysiol. 83 (2000) 1733–1750.
[6] S. Funahashi, M. Inoue, Neuronal interactions related to working memory processes in the primate
prefrontal cortex revealed by cross-correlation analysis, Cerebral Cortex 10 (2000) 535–551.
[7] J.M. Fuster, The Prefrontal Cortex: Anatomy, Physiology, and Neuropsychology of the Frontal Lobe,
Raven, New York, 1997.
[8] P.S. Goldman-Rakic, Toward a circuit model of working memory and the guidance of voluntary motor
action, in: J.C. Houk, J.L. Davis, D.G. Beiser (Eds.), Models of Information Processing in the Basal
Ganglia, MIT Press, Cambridge, 1995, pp. 131–148.
[9] D. Horn, N. Levy, I. Meilijson, E. Ruppin, Distributed synchrony in a cell assembly of spiking neurons,
Neural Networks 14 (2001) 815–824.
[10] J.E. Lisman, J.M. Fellous, X.J. Wang, A role for NMDA-receptor channels in working memory, Nat.
Neurosci. 1 (1998) 273–275.
[11] H. Markram, J. Lubke, M. Frotscher, B. Sakmann, Regulation of synaptic ecacy by coincidence of
postsynaptic Aps and EPSPs, Science 275 (1997) 213–215.
[12] J. Rubin, D.D. Lee, H. Sompolinsky, Equilibrium properties of temporally asymmetric Hebbian
plasticity, Phys. Rev. Lett. 86 (2001) 364–367.
[13] S. Song, K.D. Miller, L.F. Abbott, Competitive Hebbian learning through spike-timing-dependent
synaptic plasticity, Nat. Neurosci. 3 (2000) 919–926.
[14] R.D. Traub, R.K.S. Wong, R. Milles, H. Michelson, A model of a CA3 hippocampal pyramidal neuron
incorporating voltage-clamp data on intrinsic conductances, J. Neurophysiol. 66 (1993) 635–659.
[15] M.C. van Rossum, G.Q. Bi, G.G. Turrigiano, Stable Hebbian learning from spike timing-dependent
plasticity, J. Neurosci. 20 (2000) 8812–8821.
Katsunori Kitano received the Ph.D. degree in informatics from Kyoto University, Kyoto, Japan, in 2000.
At present, he is a research fellow of the Japan Society for the Promotion of Science.