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The Journal of Neuroscience, November 15, 2001, 21(22):8782-8788
Learning in Networks of Cortical Neurons
Goded
Shahaf and
Shimon
Marom
Department of Physiology and Biophysics, Faculty of Medicine,
Technion, Haifa, 31096 Israel
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ABSTRACT |
The results presented here demonstrate selective learning in a
network of real cortical neurons. We focally stimulate the network at a
low frequency (0.3-1 Hz) until a desired predefined response is
observed 50 ± 10 msec after a stimulus, at which point the
stimulus is stopped for 5 min. Repeated cycles of this procedure ultimately lead to the desired response being directly elicited by the
stimulus. By plotting the number of stimuli required to achieve the
target response in each cycle, we are able to generate learning curves.
Presumably, the repetitive stimulation is driving changes in the
circuit, and we are selecting for changes consistent with the
predefined desired response. To the best of our knowledge, this is the
first time learning of arbitrarily chosen tasks, in networks composed
of real cortical neurons, is demonstrated outside of the body.
Key words:
learning; multielectrode array; cultured neurons; neural
network; reward; drive reduction
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INTRODUCTION |
Learning a new behavioral task is an
exploration process that involves the formation and modulation of sets
of associations between stimuli and responses. In an effort to
understand the phenomenon of learning, two different questions are
asked. (1) What are the neural mechanisms that underlie the formation
and modulation of associations? (2) What are the principles that
underlie the selection of "appropriate" associations over
"inappropriate" ones? The nature of mechanisms underlying the
formation and modulation of associations has been the topic of intense
research. Although much is yet to be discovered, many mechanisms were
described, at various levels of neural organization, that can support
activity-dependent modification of associations between stimuli and
responses. This study addresses the second question, the principles
underlying the selection of an appropriate association during
the learning process.
Our learning experiments were performed in networks containing
10,000-50,000 cortical neurons obtained from newborn rats (Baughman et
al., 1991 ), under the assumption that the organizing principles operating at the level of neuronal populations are intrinsic to neurons
and are therefore manifested ex vivo. Such cultured cortical networks were thoroughly studied by others (Ramakers et al., 1990; Murphy et al., 1992; Maeda et al., 1995; Canepari et al., 1997; Voigt
et al., 1997; Turrigiano et al., 1998), and a substantial amount
of data has been accumulated, showing that they are structurally rich,
develop and adapt functionally and morphologically over a broad range
of time scales, and are experimentally stable over weeks.
In what follows, we show that the large random cortical networks
developing ex vivo display general properties required from neural systems capable of learning: namely, numerous connections, stability of connections, and modifiability by external stimuli. We
then describe closed-loop experiments in which these biological networks interact with a computer-controlled environment and
demonstrate a simple procedure for learning and memorizing arbitrarily
chosen tasks defined in terms of neuronal firing patterns.
Specifically, we show that, during regular low-frequency stimulation,
the network explores a large space of possible connections and can be
instructed to select and stabilize one or a subset of them by
withdrawing the stimulus at the point that the connection is observed.
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MATERIALS AND METHODS |
Culture techniques. Cortical neurons are obtained
from newborn rats within 24 hr from birth, following standard
procedures. The cortex tissue is digested enzymatically and
mechanically dissociated. The neurons are plated directly onto
substrate-integrated multielectrode array (MEA) dishes (see below). The
cultures are bathed in MEM supplemented with 5% heat-inactivated horse
serum, 0.5 mM glutamine, 20 mM
glucose, and 10 mg/ml gentamycin, and maintained in an atmosphere of
37°C, 5% CO2 and 95% air in a tissue culture
incubator and during the recording phases. Half of the medium is
exchanged twice per week. Experiments are performed in the third week
after plating, thus allowing complete maturation of the neurons.
Networks that did not respond (in the third week after plating) to
repeated low-frequency stimulation (1, 0.5, and 0.3 Hz) were not kept
for additional experimentation.
The electrical activity of the cultured network is dependent on
synaptic transmission; there are many published reports (Maeda et al.,
1995; Turrigiano et al., 1998; and references therein) showing that the
electrical activity in a cultured cortical network may be blocked by
perfusion with the NMDA receptor antagonist D-2-amino-5-phosphonovalerate (APV) and the non-NMDA
receptor antagonist CNQX. We repeated these experiments using
intracellular recordings, as well as MEA recordings. We find that, in
the presence of a mixture of synaptic blockers containing 5 µM bicuculline, 10 µM DNQX, and 20 µM APV, spiking activity within the cultured network is
completely abolished (Tal, 2000).
Electrophysiological methods. We used the substrate-embedded
multielectrode array technology (see Fig. 1A)
(Gross, 1994; Meister et al., 1994). We used arrays of 60 Ti/Au/TiN
electrodes, 30 µm in diameter, and spaced 200 µm from each other
[MultiChannelSystems (MCS), Reutlingen, Germany]. The insulation
layer (silicon nitride) was pretreated with
poly-L-lysine, forming a good surface for network
development. A commercial 60 channel amplifier (B-MEA-1060; MCS) with
frequency limits of 10-3000 Hz and a gain of 1024× was used. The
B-MEA-1060 was connected to MCPPlus filter amplifiers (Alpha Omega,
Nazareth, Israel) for additional amplification (10-20×). Stimulation
through the MEA is performed using a dedicated eight channel stimulus
generator (MCS). The microincubation environment was arranged to
support long-term recordings from MEA dishes. This was achieved by
streaming a filtered, heated, and 95% humidified air-5%
CO2 gas mixture and by electrically heating the
MEA platform to 37°C. Data were digitized using two 5200a/526
analog-to-digital boards (Microstar Laboratories, Bellevue, WA). Each
channel is sampled at a frequency of 24,000 samples/sec and
prepared for analysis using the AlphaMap interface (Alpha Omega).
Spike detection. Thresholds (8× root mean square
units; typically in the range of 10-20 µV) are separately
defined for each of the recording channels before the beginning of the
experiment. No additional spike sorting techniques are applied for the
following reason. Much of our data and their interpretation correspond
to the time scale of intraburst activities. Whole-cell recordings from
single cortical neurons, both in our hands as well as in many published
records, show that the shape of an action potential changes
dramatically within bursts of activity because of the dynamics
of membrane excitability. Consequently, any attempt to sort the spikes
according to their shapes within this time scale is doomed a priori. We
were therefore forced to resort to a stricter approach that defines
elementary activities on the basis of their participation in
statistically significant activity pairs as explained below, in which
every threshold crossing is considered in the analysis. The major
limitation of this approach is that it takes more occurrences of a
particular pair to define statistical significance. This limitation was
overcome by performing long experiments.
Definition of activity pairs. We operationally define pairs
of neural connectivity in terms of an action potential A that is
followed by another action potential B, with a precise time delay ( ±0.5 msec) between the two. A and B may be action potentials recorded
in the same or in different measuring electrodes. Both events (A and B)
are defined by threshold crossing as explained above. The number of
measuring electrodes (Ne) dictates the maximal number of
detectable pairs. Thus, for a > 0, the maximal number of
A B pairs is Ne2. For = 0, the
maximal number of AB pairs is Ne(Ne  1); an activity cannot pair with itself within a zero time delay.
Statistical significance of activity pairs. The statistical
significance (p value) of a given AB pair is
calculated under binomial distribution assumptions given the number of
times A occurred, the number of times AB occurred with a time delay
, and the probability of event B. Thus, if
p(k) is the probability of observing k
or more AB pairs out of n A events, and
pB is the probability of a B event,
then:
p < 0.01 was used as a significance measure.
Functional strength of activity pairs. Given an AB
activity pair, the forecasting of B by A, which is the strength of the functional connectivity between the two, is given in terms of a
correlation coefficient, calculated from the number of times that the
given pair appears within 1 hr, divided by the number of occurrences of
A or B (see Fig. 1B, inset).
Stability of activity pairs. For each AB pair,
statistical significance of a change in pair co-occurrence counts was
calculated under the assumptions of the binomial distribution (see
above and Fig. 1C). For instance, suppose that AB pair
(e.g., with = 20 msec) appeared
n1 times in the first 0.5 hr bin and
n4 times in the fourth 0.5 hr bin. To
state that n4 is significantly different from n1, we calculate the
probability of finding n4 or more
events (for the case of n4 > n1) or
n4 or less events (for the case of
n4 < n1). This is done using the frequency
of AB (at = 20 msec) in the first 0.5 hr bin as the
theoretical probability and the number of A events in the fourth 0.5 hr
bin as the number of trials. If the calculated probability is
p < 0.01, then n4 is
significantly different from n1.
Stimulation parameters. The pair of stimulating electrodes
was chosen according to its ability to induce a reverberating
electrical activity of the type shown in Figure 1, D and
E, in response to a biphasic current pulse (±50 µA or
smaller, lasting <500 µsec; 250 µsec for each phase). At
stimulation frequencies higher than 1 Hz, the networks usually
inactivate after a few pulses. Therefore, in the learning experiments,
the frequency of stimulation for a given network was set at either 1, 0.5, or 0.3 Hz, the highest that was possible for the particular
network without inactivating its response (see Figs. 2-6). Stimulating
electrodes were spatially near each other (~200-400 µm apart).
Peristimulus time histogram construction. A
series of 1200 stimuli (420 µsec, 50 µA, 0.3 Hz) was delivered
through a pair of electrodes, and the responses in 10 randomly chosen
active electrodes were recorded (see Fig. 1E). The
total number of responses (counted in 1 msec time bins) divided by
12000 is presented, time-locked to the stimulus event.
Stimulation protocol and analysis of activity-dependent change of
activity pairs. Each network was exposed to nine stimulation sessions (see Fig. 1F). The pattern of stimulation in
each of these nine sessions was one of the following: pattern 1, 10 min at 0.3 Hz; pattern 2, 2 min at 0.3 Hz, followed by 8 min of no stimulation; or pattern 3, 10 min of no stimulation. Each of these stimulation patterns was delivered three times to each network, in a
random temporal order. Every stimulation session was preceded and
succeeded by 100 test stimuli. For a given network, all the stimuli,
including the test stimuli, were delivered through the same pair of
electrodes. The test stimuli enabled us to define significantly
occurring activity pairs as explained above, with the prefix of each
pair (A in AB) being the stimulus itself. Using the binomial theorem
(see above and Fig. 1B,C), we
identified activity pairs whose count changed during stimulation
patterns 1 and 2 in a statistically significant manner and display the average number of such pairs normalized to the average spontaneous change (pattern 3). The data shown in Figure 1F was
obtained from testing all AB pairs (A being the stimulus itself)
with pair time delay () between 0 and 100 msec, in 1 msec bins.
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RESULTS |
The cultured neurons form numerous synaptic connections. This is
apparent from the large number of statistically significant correlated
activities between pairs of electrodes. We operationally define such
pairs of neural connectivity in terms of an action potential A that is
followed by another action potential B with a precise time delay ( ±0.5 msec) between the two (see Materials and Methods). Analysis of
the spontaneous activity of the network, without any stimulation,
suggests that the average number of such statistically significant
AB connections is a large percentage of the maximum that is possible
at relatively small values of (Fig.
1B). As the time delay
between the activities of the elements of the pair becomes longer, the
realized number of pairs decreases. Of course, a significant occurrence
of AB connection might represent a causal relationship between the
activity of A and that of B or a noncausal correlation resulting from
coactivation by a common source. Furthermore, many of the observed
connections are actually parts of larger groups of significantly
connected activities. However, as we proceed, it will become clear
that, for the purposes of this study, distinctions between the
possibilities mentioned above are not crucial. Rather, the important
thing is that the number of connections is large (Fig.
1B) and that many independent activity patterns
exist. The latter is implied from the fact that, in these networks,
single neurons seldom fire spontaneously without being activated by
other neurons (Maeda et al., 1995; Canepari et al., 1997) (see
Materials and Methods), whereas the average correlation between
elements of pairs is rather weak (Fig. 1B, inset). The stability of connections in the network may be
appreciated by comparing the number of times each of the significantly
occurring pairs appeared in 10 consecutive time bins (30 min each) over 5 hr of continuous recording of spontaneous activity, without any
stimulation. We used the number of times that a given AB activity
pair appeared in the first 30 min bin, divided by the number of
occurrences of A or B as a measure for the occurrence probability of a
pair. Using the binomial theorem, we identified pairs whose
count did not change in a statistically significant manner in
subsequent time bins. Figure 1C shows that ~70% of the pairs remained unchanged after 5 hr of spontaneous activity.
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Figure 1.
A, Large random cortical networks
cultured on substrate-embedded multielectrode arrays. Scale bar, 30 µm. B, The average number (4 networks) of
significantly occurring activity pairs formed between 10 randomly
chosen active (>0.2 Hz of spontaneous activity) electrodes. (Data are
obtained from spontaneous activity, without any stimulation.) This
number, normalized to the maximal number of possible activity pairs, is
depicted as fraction connected; depicts the within-pair time delay.
Inset, Average functional connectivity strength as a
function of (4 different networks; 10 randomly chosen active
electrodes from each). C, Stability of neural activity
pairs over hours. Ten active (>0.2 Hz of spontaneous activity)
electrodes were chosen randomly. All of the occurrences of pairs
( = 50 msec) (spontaneous activity, without any stimulation)
are counted in 10 0.5 hr bins. For each such time bin, the
corresponding point in the graph shows the fraction of
pairs that their count did not change in a statistically significant
manner (p < 0.01) relative to their count
in the first 0.5 hr. Data were averaged over four networks, and
SD bars were added. D, A stimulus pulse lasts 420 µsec, and its amplitude is 50 µA. The traces were recorded
simultaneously from different electrodes. E,
Peristimulus time histogram. The first peak represents
direct activation of neurons by the stimulus; the second
peak represents a reverberating response. F,
Stimulation-induced change in occurrence of activity pairs (average of
4 networks; data are normalized to spontaneous change; for details, see
Materials and Methods). Note that, as the stimulation series become
longer, more activity pairs change (increased or decreased) the number
of their occurrences. Interestingly (as can be understood from SD error
bars), it is possible to find networks in which the short pattern (2 min stimuli) stabilizes pairs compared with spontaneous change with
time.
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When stimulating currents are delivered through a pair of
substrate-embedded electrodes at a constant frequency, the network responds by generating a rich repertoire of reverberating electrical activities, lasting 100 msec or more (Fig.
1D,E). Modifications in functional
connectivity would be manifested as changes in the coupling of such
responses to the stimulus. Indeed, repeated stimulation induces changes
in network responsiveness, as shown previously by others (Jimbo et al.,
1999). Furthermore, Figure 1F shows that the
magnitude of such modifications increases with stimulation time,
reflecting the myriad activation pathways and activity-dependent mechanisms that operate in these networks. This "exploratory" nature (of the change in response to series of stimuli) is further demonstrated in the data presented below.
The analyses presented above imply that cortical networks display
general properties expected from neural systems capable of learning:
namely, numerous connections, stability of connections, and
modifiability by external stimuli. We now turn to the novel aspect of
this study, which is demonstrating learning in a cortical network
without the involvement of a neural rewarding entity. The idea is
simply to stimulate the network until the required response is
attained, and once this occurs, to remove the "driving" stimulus.
We then ask how long it takes to attain the required response. Will the
appropriate responsiveness remain stable after such a simple procedure?
How selective can such a change in connectivity be? If after the
procedure the required response to stimulus occurs reliably and
selectively, this could be considered as a form of learning.
Each experiment starts by stimulating the network through a pair of
electrodes and observing the responsiveness of all other (i.e., the
nonstimulated) electrodes. A nonstimulated electrode that responds
50 ± 10 msec after a stimulus with a response-to-stimulus (R/S)
ratio of 1/10 or less is selected. In other words, before training, it
takes at least 10 stimuli to evoke one action potential in the selected
electrode within the designated time frame of 50 ± 10 msec after
a stimulus. During the training phase, the learning task is to increase
the R/S of the selected electrode to 2/10 or greater at the designated
time window of 50 ± 10 msec after a stimulus. The two stimulated
electrodes are continuously stimulated at a constant frequency of 1/3,
1/2, or 1 stimulus per second. A computer constantly monitors the R/S
of the selected electrode, and once the criterion of R/S  2/10
is fulfilled, i.e., whenever two responses were seen in any 10 consecutive trials, the computer automatically stops the stimulation.
After 5 min, the network is stimulated again (at the same low
frequency) until the criterion R/S 2/10 in the same selected
electrode is fulfilled again. This stimulation cycle, which is composed
of 5 min without stimulation followed by low-frequency (0.3, 0.5, or 1 Hz) stimulation until R/S 2/10 criterion in the selected
electrode is fulfilled, is repeated many times. As a rule, if the
criterion is not fulfilled within 10 min of stimulation, the
stimulation is stopped for 5 min. Hence, the maximal duration of one
stimulation cycle is 15 min (i.e., 10 min of stimulation and 5 min of
quiescence). The latency for reaching the predetermined criterion
(referred to as response time) in each stimulation cycle is used as a
measure for the strength of S-R connection and may be viewed as a
measure of the degree to which the task was learned.
An example for the result of this learning procedure is shown in Figure
2. It includes the responses of a
selected electrode before (left column) and after
(right column) training. The 11 traces of each
panel show the responses to 11 consecutive stimulation pulses. Note that the activity within the 50 ± 10 msec window (depicted) is markedly increased after the training phase.
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Figure 2.
Example of learning in a cultured network of
cortical neurons. Each trace within a
panel shows recordings obtained 10 msec before the
stimulus to 70 msec after the stimulus, before (left)
and after (right) the training procedure. Note that the
responsiveness of the electrode within the designated time window
increased appreciably.
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Figure 3A shows eight learning
curves, differing in the learning kinetics. In these curves, the
response time (i.e., time required for the selected electrode to
fulfill the R/S 2/10 criterion) is plotted against the number
of stimulation cycle. (Recall that each stimulation cycle is composed
of 5 min without stimulation, followed by low-frequency stimulation
until R/S 2/10 criterion is fulfilled.) The curves are
characterized by response time decrement and stabilization at lower
values compared with the initial values. Note that the time required to
instruct a network to perform the task varies, reflecting the
arbitrariness of the procedure by which the tasks are chosen and the
idiosyncrasies of the networks.
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Figure 3.
A, Eight learning curves, differing
in their learning kinetics. The response time (i.e., the duration of
stimuli series until criterion is fulfilled) is plotted against the
number of stimulation cycle. Each point depicts the time
(in seconds) to accomplish the task in one cycle. B,
Four control curves from a protocol in which each cycle consisted of 10 min of stimulation and 5 min of quiescence, regardless of response (see
Results). Vertical broken lines are referred to
in Figure 5.
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The notion that driving stimulus removal is necessary for selecting
appropriate network responses is further supported by a control
experiment. In this experiment, the fulfillment of the R/S criterion in
the selected electrode did not lead to stimulus removal (i.e., the
attainment of the criterion was ignored). The stimulation was delivered
for 10 min interrupted by 5 min of quiescence, regardless of the
responses recorded from the selected electrode. Figure 3B
shows that, under these conditions, the response time (i.e., the time
required for first appearance of R/S 2/10 within each
stimulation cycle) plotted against the stimulation cycle number shows
large fluctuations and a tendency to a decreased responsiveness over time.
Thus far, our criterion for stopping the stimulus has simply been the
appearance of a response on the selected electrode. We refer to the
eight trials that used this criterion as "simple learning" trials.
To ensure selectivity of the R/S increase in the selected electrode, we
also conducted 16 trials using a second criterion. In these trials, we
monitored a second electrode in the array, which serves as a measure
for global network responsiveness. Our condition for removing the
stimulus was that the R/S criterion be fulfilled in the selected
electrode and not fulfilled in the second monitored electrode. We refer
to these as "selective learning trials." Of these 16 selective
learning trials, eight showed learning (Fig.
4). In the remaining eight selective
learning trials, the latency for reaching the predetermined R/S
criterion did not relax; that is, the response times did not decrease
and did not stabilize at lower values compared with the initial values.
Such "nonrelaxing" experiments were stopped after 25 stimulation
cycles.
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Figure 4.
Eight selective learning curves, differing in
their learning kinetics. The response time (i.e., the duration of
stimuli series until criterion is fulfilled) is plotted against the
number of stimulation cycle. Each point depicts the time
(in seconds) to accomplish the task in one cycle. Vertical
broken lines are referred to in Figure 5.
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Figure 5 (left eight columns)
summarizes the selective learning data. Changes in R/S of the selected
electrodes (filled circles) and 10 control electrodes
(stars) are depicted for eight experiments from eight
different networks. For each network, the 10 control electrodes were
chosen by analyzing the data, after the completion of the experiment,
based on their similarity to the R/S of the selected electrode before
the training; specifically, the control electrodes are the 10 electrodes whose R/S before training were the most similar to the R/S
of the selected electrode before training. The change, depicted by
f, is defined as the ratio between the responsiveness before
training and responsiveness after training, normalized to the change in
R/S of the selected electrode. Thus, f = 1 means
a change in R/S that is identical to the change measured in the
selected electrode. f > 1 and f < 1 mean that the relevant response of a control electrode increased or
decreased, respectively, relative to the selected electrode. Note that
the strengthening in the response-to-stimulus ratio (R/S) of the
selected electrode is generally higher relative to the responsiveness
change in the control electrodes. Also note that, because the selected
and control electrodes demonstrate low responsiveness before the
training, a bias toward an average increase of R/S during training is
introduced. The reported effect is selective because the increase in
R/S of the selected electrode is more than the average increase for the control electrodes. The probability of the selected electrode to be
ranked fourth or higher (of 11), as is the case in the eight experiments shown, is  (4/11)8. Note
that, in the control trials (four right columns), no
preferred ranking of the selected electrode is observed.
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Figure 5.
Changes in R/S, depicted by
f, of the selected electrode (filled
circles) and 10 control electrodes (stars) for
eight selective learning experiments (left) and four
control experiments (right) whose curves are shown in
Figure 3B. The left eight columns
correspond to the eight selective learning curves of Figure 4 by a
left-to-right,
top-to-bottom order. (For instance, the
selectivity data that relates to the top right curve of
Fig. 4 is shown in the second column from the
left in this figure). For the purpose of calculating
f, the point in time that separates the period before
training and that of after training is depicted by broken
lines in the curves of Figures 3B
and 4. f is normalized to the R/S change of the selected
electrode.
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Figure 6 summarizes the
entire data set obtained in the above described experiments: the
average control curve (curve 1), the average behavior of the
entire set of trials (curve 2; including the
nonrelaxing trials), and the average learning curve (curve 3; the combined set of simple and selective learning curves). Each
point depicts the average time (in seconds) to accomplish the task in one cycle within a series of cycles. Figure 6 provides an
indication for the robustness of the main phenomena shown in this
study: when the loop is closed and the response is allowed to remove
the stimulus, learning curves may be obtained; when the loop is open,
i.e., the computer is instructed not to remove the stimulus when the
selected electrode criterion is fulfilled, the curves "explore
away."
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Figure 6.
Average control (curve 1), average
response over all noncontrol trials (curve 2), and
average learning curve (curve 3). Each
point depicts the average time (in seconds) to
accomplish the task in one cycle. Filled gray circles
(in curve 2) and filled black squares (in
curve 3) depict points that are significantly different
from averaged control (curve 1; F test,
p < 0.05). Fitted power function lines
(y = axb) are added to
emphasize the different trends of the curves. Note that this figure
shows averages of trials with different durations (curves from Figs. 3
and 4, as well as eight nonrelaxing curves that were interrupted after
25 cycles; see Results). Hence, the number of samples
contributing to each point varies as follows: curve 1,
n = 4 for cycles 1-40; curve 2,
n = 24 for cycles 1-10, n = 16 for cycles 11-20, and n = 8 for cycles 21-25;
curve 3, n = 16 for cycles 1-10,
n = 8 for cycles 11-20, and n = 4 for cycles 21-40.
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DISCUSSION |
The experiments described above show that sufficient conditions
for the realization of learning by a selection process, without the
involvement of a neural rewarding entity, are embodied in large random
networks of neurons maintained ex vivo. These networks form
a large space of connectivity configurations that are stable over many
hours. The connectivity can be modulated by external focal stimulation
in an activity-dependent manner. Most importantly, the networks explore
the space of possible responses and stabilize at configurations that
remove the stimuli.
From the theoretical point of view, the above demonstration conveys an
important message, supported by behavioral studies and psychological
theories advocated over 50 years ago by eminent psychologists such as
Hull (1943) and Guthrie (1946): it is not necessary to assume a
separate mechanism for the biological realization of a reward in
distinction from the process of exploration for solutions; the
behavioral concept of reward might be considered as a change (removal)
in the drive underlying the exploration in the space of possible modes
of network response. A drive to explore that is removed when a desired
state is achieved is an intentionless natural principle for adaptation
to rich and unlabeled environment. Of course, the fact that learning by
stimulus removal is plausible biologically does not mean that it is
implemented in real brains; however, the simplicity of this principle
makes it very likely that it does.
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FOOTNOTES |
Received July 24, 2001; revised Sept. 4, 2001; accepted Sept. 5, 2001.
This study is supported by grants from the European Commission, the
Israel Science Foundation, and the Bernard Katz Minerva Foundation. We
thank Vladimir Lyakhov and Ella Romanko-Lyakhov for their invaluable
technical support and Dr. Noam Ziv for encouragement and helpful
discussions. We also thank Drs. Shraga Hocherman, Larry Manevitz,
Daniel Dagan, Itzik Schiller, and Jackie Schiller for their suggestions
in the preparation of this manuscript.
Correspondence should be addressed to Shimon Marom, Department of
Physiology and Biophysics, Faculty of Medicine, Technion, Haifa, 31096 Israel. E-mail: marom{at}tx.technion.ac.il.
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