Self-organization of Hebbian Synapses 
in Hippocampal Neurons 
Thomas H. Brown, 
Zachary F. Mainen,* 
Department of Psychology 
Yale University 
New Haven, CT 06511 
Anthony M. Zador,* and Brenda J. Claiborne* 
Division of Life Sciences 
University of Texas 
San Antonio, TX 78285 
ABSTRACT 
We are exploring the significance of biological complexity for neuronal 
computation. Here we demonstrate that Hebbian synapses in realistical- 
ly-modeled hippocampal pyramidal cells may give rise to two novel 
forms of self-organization in response to structured synaptic input. First, 
on the basis of the electrotonic relationships between synaptic contacts, 
a cell may become tuned to a small subset of its input space. Second, the 
same mechanisms may produce clusters of potentiated synapses across 
the space of the dendrites. The latter type of self-organization may be 
functionally significant in the presence of nonlinear dendritic conduc- 
tanceso 
1 INTRODUCTION 
Long-term potentiation (LTP) is an experimentally observed form of synaptic plasticity 
that has been interpreted as an instance of a Hebbian modification (Kelso et al, 1986; 
Brown et al, 1990). The induction of LTP requires synchronous presynaptic activity and 
postsynaptic alepolarization (Kelso et al, 1986). We have previously developed a detailed 
biophysical model of the LTP observed at synapses onto hippocampal region CA1 pyrami- 
39 
40 Brown, Mainen, Zador, and Claiborne 
Figure 1: Two-dimensional projection of a reconstructed hippocampal CA1 pyramidal cell. 
dal neurons (Zaclor et al, 1990). The synapses at which this tbrm of LTP occurs are distrib- 
uted across an extensive dendritic arbor (Fig. 1). During synaptic stimulation, the 
membrane voltage at each synapse is different. In this way, a biological neuron differs 
from the processing elements typically used in neural network models, where the postsyn- 
apfic activity can be represented by a single state variable. We have developed an electro- 
tonic model based on an anatomically reconstructed neuron. We have used this model to 
explore how the spatial distribution of inputs and the temporal relationships of their activa- 
tion affect synapfic potentiation. 
2 THE NEURONAL MODEL 
Standard compartmental modeling techniques were used to represent the electrical struc- 
ture of hippocampal CA1 pyramidal cells. 
2.1 MORPHOLOGY AND ELECTRICAL PARAMETERS 
Morphometric data were obtained from three-dimensional reconstructions (Brown et al., 
1991) of hippocampal neurons (Fig. 1). A correction factor was applied to the membrane 
area based on an estimate for spine density of 2 / grn. The original measurements divided 
a single neuron into 3000-4000 cylinders with an average length of 5.5 grn. For simulation 
purposes, this structure was collapsed into 300-400 compartments, preserving the connec- 
tivity pattern and changes in process diameter. Electrical constants were Rm = 70 ldl-crn 2, 
Cm-- 1 gF/cm 2, Ri = 200 fl-crn (Spruston & Johnston 1990). The membrane was electri- 
cally passive. Synaptic currents were modeled as the sum of fast AMPA and slow NMDA 
conductances on the head of a two-compartment spine (Zador et al., 1990). The AMPA 
conductance was represented by an alpha function (Jack et al., 1975) with time constant of 
1.5 rnsec (Brown and Johnston, 1983). The NMDA conductance was represented by a 
more complicated function with two time constants and a voltage dependence due to volt- 
age-sensitive channel blocking by Mg 2+ ions (see Zador et al., 1990; Brown et al. 1991). 
The initial peak conductances, gastt,a and gJvstoa' were set to 0.5 and 0.1 nS respectively. 
Self-organization of Hebbian Synapses in Hippocampal Neurons 41 
2.2 SIMULATION AND SYNAPTIC MODIFICATION 
Simulations were run on a Sun 4/330 workstation using a customized version of NEURON, 
a simulator developed by Michael Hines (Hines, 1989). Prior to a simulation, 5 patterns of 
40 synapses were selected at random from a pool of synapses distributed uniformly over 
the apical and basal dendrites. Simulations were divided into trials of 100 msec. At the 
beginning of each trial a particular pattern of synapses was activated synchronously (3 
stimuli at intervals of 3 msec). The sequential presentation of all 5 selected patterns 
constituted an epoch. An entire simulation consisted of 20 presentation epochs. Over the 
course of each trial, membrane potential was computed at each location in the dendritic 
tree, and these voltages were used to compute weight changes Awij according to the 
Hebbian algorithm described below. After each trial, the actual peak AMPA conductances 
(gal,,a, hereafter denoted g) were scaled by the sigmoidal function 
gmax 
g,, = e  (%- o.5,,) 
1+ 
where t determines the steepness of the sigmoid, and g was set to 1.0 nS. 
The rule for synaptic modification was based on a biophysical interpretation (Kairiss et al., 
1991; Brown et al., 1991) of a generalized bilinear form of Hebbian algorithm (Brown et 
al., 1990): 
Awij =  [ai(t),aj(t ) ] - [ai(t ) ] - [aj(t) ] -8, (2) 
where et, [, and  are functionals, $ is a constant, a..(t) represents postsynaptic activity and 
a(O represents presynaptic activity. This equation pecifies an interactive form of synaptic 
enhancement combined wth three nomnteractve forms of synaptac depression, all of 
which have possible neurobiological analogs (Brown et al, 1990). The interactive term was 
derived from a biophysical model of LTP induction in a spine (Zador et al. 1990). A sim- 
plified version of this model was used to compute the concentration of caz+-bound calm- 
odulin, [CaM-Ca4]. It has been suggested that CaM-Ca4 may trigger protein kinases 
responsible for LTP induction. In general [CaM-Ca4] was a nonlinear function of subsyn- 
aptic voltage (Zador et al., 1990). 
The biophysical mechanisms underlying synaptic depression are less well understood. The 
constant $ represents a passive decay process and was generally set to zero. The functional 
[ represents heterosynaptic depression based on postsynaptic activity. In these simula- 
tions, [ was proportional the amount of depolarization of the subsynaptic membrane from 
resting potential (Vy,, - V,.,,). The functional  represents homosynaptic depression based 
on presynaptic activity. Hre,  was proportional to the AMPA conductance, which can 
be considered a measure of exclusively presynaptic activity because it is insensitive to 
postsynaptic voltage. The three activity-dependent terms were integrated over the period 
of the trial in order to obtain a measure of weight change. Reinterpreting et, [, and  as con- 
stants, the equation is thus: 
Awij = I [tl [CamCa4] - [ (Vsyn- Vr*st) -- gaMt'a--J] dt' 
trial 
(3) 
42 Brown, Mainen, Zador, and Claiborne 
-40 
-60 
0 50 
! 
-60 ..:" ............. 
0 50 100 
-80 .... , ....  .... , .... 
0 5 10 15 20 
e.ocks 
Figure 2: Interactions among Hebbian synapses produce differing global effects ("winning" and 
"losing" patterns) on the basis of the spatial distribution of synapses. The PSP (always measured 
at the soma) due to two different patterns of 40 synapses are plotted as a function of the presentation 
epoch. Initially, pattern 1 (solid line) evoked a slightly greater PSP than pattern 2 (dotted line; in- 
set, top fight). After 20 epochs these responses were reversed: the PSP due to pattern 1 was de- 
pressed while the PSP due to pattern 2 was potentiated (inset, top left). 
3 RESULTS 
Analysis of the simulations revealed self-organization in the form of differential modifica- 
tion of synaptic strengths (Mainen et al. 1990). Two aspects of the self-organization phe- 
nomena were distinguished. In some simulations, a form of pattern selection was observed 
in which clear"winners" and "losers" emerged. In other simulations, the average synaptic 
efficacy remained about the same, but spatial heterogeneities clustering---of synaptic 
strength developed. Different measures were used to assess these phenomena. 
3.1 PATTERN SELECTION 
The change in the peak postsynaptic potential recorded at the soma (PSP) provided one use- 
ful measure of pattern selection. In many simulations, pattern selection resulted in a 
marked potentiation of the PSP due to some patterns and a depression of the PSP due to 
others. The PSP can be regarded as an indirect measure of the functional consequence of 
self-organization. In the simulation illustrated in Fig. 2, patterns of 40 synapses produced 
an average PSP of 15 mV before learning. After learning, responses ranged from 10% to 
150% of this amount Underlyinjg. pattem selection was a change in the average peak syn- 
aptic conductance for the pattemg,y,0). 1 The initial value of ,,y,, was the same for all pat- 
terns, and its final value was bounded by eq. 1. In many simulations, ,y, approached the 
upper bound for some patterns and the lower bound for other patterns (Fig. 3). In this way, 
the neuron became selectively tuned to a subset of its original set of inputs. The specificity 
Self-organization of Hebbian Synapses in Hippocampal Neurons 43 
1.0 
0 
5 10 15 2,0 
epoch, s 
Figure 3. The mean synaptic conductance snOf two patterns is plotted as a function of the pre- 
sentation epoch. Both patterns began with iaefiucal total synapfic strength (40 synapses with go,, = 
0.5 nS). Synaptic conductances were constrained to the range [0.0, 1.0] nS. After twenty epochs, 
 of pattern 1 (solid line) approached the minimum of 0.0 nS while syn of pattern 2 (dotted line) 
syn 
approached the maximum of 1.0 nS. 
of this tuning was dependent on the parameter values of the neuronal model, learning rule, 
and stimulus set. 
3.2 CLUSTER FORMATION 
Heterogeneity in the spatial distribution of strengthened and weakened synapses was often 
observed. After learning, spatial clusters of synapses with similar conductances formed. 
These spatial heterogeneities can be illustrated in several ways. In one convenient method 
(see Brown et al., 1991), synapses are represented as colored points superimposed on a ren- 
dition of the neuronal morphology as illustrated in Fig. 1. By color-coding g:y for each 
synapse in a pattern, correlations in synaptic strength across dendritic space are immediate- 
ly apparent. In a second method, better suited to the monochrome graphics available in the 
present text, the evolution of the variance of g.,y is plotted as a function of time (Fig. 4). 
In the simulation illustrated here, the increase in variance was due to the formation of a sin- 
gle, relatively large cluster of strengthened synapses. Within other parameter regimes, mul- 
tiple clusters of smaller size were formed. 
4 DISCUSSION 
The important differences between synaptic modifications in the biophysically-modeled 
neuron and those in simple processing elements arise from voltage gradients present in the 
realistic model (Brown et al., 1991; Kairiss et al., 1990). In standard processing elements, 
1 Although syn and the somatic PSP were generally correlated, the relationship between the two is 
not linear, as was often evident in simulations (compare initial trials in Figs. 2 and 3). 
44 Brown, Mainen, Zador, and Claiborne 
1.0 
% 
0.0 , 
5 10 15 2,0 
epoch, s 
Figure 4: Synapfic heterogeneity is indicated by increases in the variance (o '2) of the set of synaptic 
conductances for each pattern. The variances of the peak synapfic conductances (g,yn) of 4 patterns 
are plotted as a,___.nction of the epoch. The variance of all 4 patterns approached the theoretical 
maximum of d0.5. In this parameter regime, the variance was due to the potentiation of a single 
large cluster of synapes combined with the depression of other synapses. 
a single state variable represents postsynaptic activity. In contrast, the critical subsynaptic 
voltages which represent postsynaptic activity in the neuron are correlated but are not strict- 
ly equal. The structure and electrical properties of the cell interact with its synaptic input 
to determine the precise spatiotemporal pattern of membrane voltage. Thus, the voltage at 
any synapse depends strongly on its electrotonic relationships to other active synapses. The 
way in which this local alepolarization affects the nature of self-organization depends on the 
specific mechanisms of the synaptic modification rule. We have modeled a pair of oppos- 
ing voltage-dependent mechanisms. An interactive potentiation mechanism (the functional 
c) promotes cooperativity between spatially proximal synapses with temporally correlated 
activity. A heterosynaptic depression mechanism (the functional I]), which is independent 
of presynaptic activity, promotes competition among spatially proximal synapses. 
Through mechanisms such as these, the specific electrotonic structure of a neuron prede- 
termines a complex set of interactions between any given spatial distribution of synaptic 
inputs. We have shown that these higher-order interactions can give rise to self-organiza- 
tion with at least two interesting effects. 
4.1 SPARSE REPRESENTATION 
The phenomenon of pattern selection demonstrates how Hebbian self-organization may 
naturally tune neurons to respond to a subset of their input space. This tuning mechanism 
might allow a large field of neurons to develop a sparse coding of the activity in a set of 
input fibers, since each neuron would respond to a particular small portion of the input 
space. Sparse coding may be advantageous to associative learning and other types of neural 
computation (Kanerva, 1988). 
Self-organization of Hebbian Synapses in Hippocampal Neurons 45 
4.2 CLUSTERING AND NONLINEAR COMPUTATION 
The formation of clusters of strengthened synapses illustrates a property of Hebbian self- 
organization whose functional significance might only be appreciated in the presence of 
nonlinear (voltage-dependent) dendritic conductances. We have examined the self-organi- 
zation process in an electrically passive neuron. Under these conditions, the presence of 
clustering within patterns has litfie effect on the observed output. In fact, it is known that 
hippocampal cells of the type modeled possess a variety of spatially heterogeneous nonlin- 
ear dendritic conductances (Jones et al., 1989). The computational role of such nonlinear- 
ities is just beginning to be explored. It is possible that interactions between synaptic 
clustering and nonlinear membrane patches may significantly affect both the performance 
of dendritic computations and the process of self-organization itself. 
Acknowledgments 
This research was supported by grants from the Office of Naval Research, the Defense Ad- 
vanced Research Projects Agency, and the Air Force Office of Scientific Research. 
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