Nonlinear Pattern Separation in Single Hippocampal 
Neurons with Active Dendritic Membrane 
Anthony M. Zador t Brenda J. Claiborne  Thomas H. Brown t 
*Depts. of Psychology and Cellular 
& Molecular Physiology 
Yale University 
New Haven, CT 06511 
zadoryale.edu 
Division of Life Sciences 
University of Texas 
San Antonio, TX 78285 
ABSTRACT 
The dendritic trees of cortical pyramidal neurons seem ideally suited to 
perform local processing on inputs. To explore some of the implications 
of this complexity for the computational power of neurons, we simulated 
a realistic biophysical model of a hippocampal pyramidal cell in which a 
"cold spot"--a high density patch of inhibitory Ca-dependent K channels 
and a colocalized patch of Ca channels--was present at a dendritic 
branch point. The cold spot induced a nonmonotonic relationship be- 
tween the strength of the synaptic input and the probability of neuronal 
fn-ing. This effect could also be interpreted as an analog stochastic XOR. 
1 INTRODUCTION 
Cortical neurons consist of a highly branched dendritic tree that is electrically coupled to 
the soma. In a typical hippocampal pyramidal cell, over 10,000 excitatory synaptic inputs 
are distributed across the tree (Brown and Zador, 1990). Synaptic activity results in current 
flow through a transient conductance increase at the point of synaptic contact with the 
membrane. Since the primary means of rapid intraneuronal signalling is electrical, infor- 
mation flow can be characterized in terms of the electrical circuit defined by the synapses, 
the dendritic tree, and the soma. 
Over a dozen nonlinear membrane channels have been described in hippocampal pyrami- 
dal neurons (Brown and Zador, 1990). There is experimental evidence for a heterogeneous 
distribution of some of these channels in the dendritic tree (e.g. Jones et al., 1989). In the 
absence of these dendritic channels, the input-output function can sometimes be reasonably 
approximated by a modified sigmoidal model. Here we report that introducing a cold spot 
51 
52 Zador, Claiborne, and Brown 
at the junction of two dendritic branches can result in a fundamentally different, nonmono- 
tonic input-output function. 
2 MODEL 
The biophysical details of the circuit class defined by dendritic trees have been well char- 
acterized (reviewed in Rall, 1977; Jack et al., 1983). The fundamental circuit consists of a 
linear and a nonlinear component. The linear component can be approximated by a set of 
electrical compartments coupled in series (Fig. 1C), each consisting of a resistor and capac- 
itor in parallel (Fig. lB). The nonlinear component consists of a set of nonlinear resistors 
in parallel with the capacitance. 
The model is summarized in Fig. 1A. Briefly, simulations were performed on a 
3000-compartment anatomical reconstruction of a region CA1 hippocampal neuron (Clai- 
borne et al., 1992; Brown et al., 1992). All dendritic membrane was passive, except at the 
cold spot (Fig. 1A). At the soma, fast K and Na channels (cf. Hodgkin-Huxley, 1952) gen- 
erated action potentials in response to stimuli. The parameters for these channels were 
modified from Lytton and Sejnowski (1991; cf. Borg-Graham, 1991). 
A 
S  
Synaptic 
input 
Cold spot 
Fast somatic 
and Na curren 
B 
Om gNa gca gsyn 'gL 
-ENaTEoaTEsy n TEK TEL 
<  
Radial and longitudinal Ca +2 diffusion 
Fig. 1 The model. (A) The 3000-compartment electrical model used in these simulations was ob- 
tained from a 3-dimensional reconstruction of a hippocampal region CA1 pyramidal neuron (Clai- 
borne et al, 1992). Each synaptic pathway (A-D) consisted of an adjustable number of synapses 
at'rayed along the single branch indicated (see text). Random background activity was generated with 
a spatially uniform distribution of synapses firing according to Poisson statistics. The neuronal mem- 
brane was completely passive (linear), except at the indicated cold spot and at the soma. (B) In the 
nonlinear circuit associated with a patch a neuronal membrane containing active channels, each chan- 
nel is described by a voltage-dependent conductance in series with its an ionic bauer), (see text). In 
the present model the channels were spatially localized, so no single patch contained all of the non- 
linearities depicted in this hypothetical illustration. (i+ A dendritic segment is illustrated in which 
both electrical and Ca 2+ dynamics were modelled. C buffering, and both radial and longitudinal 
Ca 2+ diffusion were simulated. 
Nonlinear Pattern Separation in Single Hippocampal Neurons 53 
We distinguished four synapfic pathways AoD (see Fig. 1A). Each pathway consisted of a 
population of synapses activated synchronously. The synapses were of the fast AMPA type 
(see Brown et. al., 1992). In addition, random background synapfic activity distributed uni- 
formly across the dendrific tree fired according to Poisson statistics. 
The cold spot consisted of a high density of a Ca-activated K channel, the AHP current 
(Lancaster and Nicoll, 1987; Lancaster et. al., 1991) colocalized with a low density patch 
of N-type Ca channels (Lytton and Sejnowski, 1991; cf. Borg:Graham, 1991). Upon local- 
ized depolarizafion in the region of the cold spot, influx of ca ':+ through e Ca channel re- 
sulted in a transient increase in the local [ca':+]. The model included ca z+ buffering, and 
both radial and longitudinal diffusion in the region of the cold spot. The increased [Ca 2+] 
activated the inhibitory AHP current. The interplay between the direct excitatory effect of 
synapfic input, and its inhibitory effect via the AHP channels formed the functional basis 
of the cold spot. 
3 RESULTS 
3.1 DYNAMIC BEHAVIOR 
Representative behavior of the model is illustrated in Fig. 2. The somatic potential is plot- 
ted as a function of time in a series of simulations in which the number of activated syn- 
apses in pathway A/B was increased from 0 to about 100. For the first 100 msec of each 
simulation, background synaptic activity generated a noisy baseline. At t = 100 msec, the 
indicated number of synapses fired synchronously five times at 100 Hz. Since the back- 
ground activity was noisy, the outcome of the each simulation was a random process. 
The key effect of the cold spot was to impose a limit on the maximum stimulus amplitude 
that caused fh-ing, resulting in a window of stimulus strengths that triggered an action po- 
tential. In the absence of the cold spot a greater synaptic stimulus invariably increased the 
likelihood that a spike Fired. This limit resulted from the relative magnitude of the AHP 
srple Sometic VoltSge TraOmm 
o 
o 
Fig. 2 Sample rims. The membrane voltage at the soma is ploued as a function of time and synaptic 
stimulus intensity. At t = 100 msec, a synapfic stimulus consisting of 5 pulses was acfivitated. The 
noisy baseline resulted from random synapfic input. A single action potential resulted for input in- 
tensifies within a range determined by the kinetics of the cold spot. 
54 Zador, Claiborne, and Brown 
current "threshold" to the threshold for somatic spiking. The AHP current required a rela- 
tively high level of activity for its activation. This AHP current "threshold" reflected the 
sigmoidal voltage dependence of N-type Ca current activation (V/2 = -28 rnV), since only 
as the dendritic voltage approached V/2 did dendrific [Ca 2+] rise enough to activate the 
AHP current. Because V/2 was much higher than the threshold for somatic spiking (about 
-55 rnV under current clamp), there was a window of stimulus strengths sufficient to trigger 
a somatic action potential but insufficient to activate the AHP current. Only within this 
window of between about 20 and 60 synapses (Fig. 2) did an action potential occur. 
3.2 LOCAL NON-MONOTONIC RESPONSE FUNCTION 
Because the background activity was random, the outcome of each simulation (e.g. Fig. 2) 
represented a sample of a random process. This random process can be used to define many 
different random variables. One variable of interest is whether a spike fired in response to 
a stimulus. Although this measure ignores the dynamic nature of neuronal activity, it was 
still relatively informative because in these simulations no more than one spike fired per 
experiment. 
Fig. 3A shows the dependence of firing probability on stimulus strength. It was obtained 
by averaging over a population of simulations of the type illustrated in Fig. 2. In the ab- 
sence of AHP current (dotted line), the f'Lring probability was a sigmoidal function of activ- 
ity. In its presence, the f'u-'ing probability was a smooth nonmonotonic function of the 
activity (solid line). The fu'ing probability was maximum at about 35 synapses, and oc- 
curred only in the range between about 10 and 80 synapses. The statistics illustrated in Fig. 
3A quantfly the nonmonotonicity that is implied by the single sample shown in Fig. 2. 
Spikes required the somatic Hodgkin-Huxley-like Na and K channels. To a first approxi- 
marion, the effect of these channels was to convert a continuous variable the somatic volt- 
age into a discrete variable the presence or absence of a spike. Although this 
approximation ignores the complex interactions between the soma and the cold spot, it is 
useful for a qualitative analysis. The nonmonotonic dependence of somatic activity on syn- 
A B 
1.0 ....................... -6 ....................... 
 **  --- Cold Spot- 
 I 7   --58 Cold Spot+ 
 0.8 ___ Cold Spot-  
  Cold Spot+  
  -62 
 0.4  -64 
 c 0.2 
o.o ....... ..... -os ....................... 
0 20 40 60 80 100 120 0 20 40 60 0 100 120 
Number of active synpases Nber of active synpnses 
Fig. 3 Nomonomffic put-ouut relation. (A) Eh t rresmB e probi at at let 
one spe w Ced at e dic fivi level. h e absce of a ld  e Cg probabil- 
i creed shyly d monomcy  e nmber of sapses  paway CID e (dot. 
d ne).  cont, e Cg probabi reached a mmm for pawayA/B d en dre 
(so line). (B) Each pot rrenB e pe somatic volge for a sgle sulafion  e - 
card activi level  e presmce athway B; o le) d smce athway C/D; doed 
ne) of a cold sC Because each t reesenB e oumome of a sgle sulafio  mnt 
m e average us  (A), e  reflect e vice due m e rdom backd fivi. 
Nonlinear Pattern Separation in Single Hippocampal Neurons 55 
aptic activity was preserved even when active channels at the soma were eliminated (Fig. 
3B). This result emphasizes that the critical nonlinearity was the cold spot itself. 
3.3 NONLINEAR PATTERN SEPARATION 
So far, we have treated the output as a function of a scalarrathe total activity in pathway 
A/B (or C/D). In Fig. 3 for example, the total activity was defined as the sum of the activ- 
ities in pathway A and B. The spatial organization of the afferents onto 2 pairs of branch- 
esmA & B and C & D (Fig. 1)--suggested considering the output as a function of the 
activity in the separate elements of each pair. 
The effect of the cold spot can be viewed in terms of the dependence of f'ldng as a function 
of separate activity in pathways A and B (Fig. 4). Each filled circle indicates that the neuron 
fared for the indicated input intensity of pathways A and B, while a small dot indicates that 
it did not fare. As suggested by (Fig. 3), the f'ldng probability was highest when the total 
activity in the two pathways was at some intermediate level. The neuron did not fare when 
the total activity in the two pathways was too large or too small. In the absence of the cold 
spot, only a minimum activity level was required. 
In our model the probability of firing was a continuous function of the inputs. In the pres- 
ence of the dendritic cold spot, the comers of this function suggested the logical operation 
XOR. The probability of f'ldng was high if only one input was activated and low if both or 
neither was activated. 
4 DISCUSSION 
4.1 ASSUMPTIONS 
Neuronal morphology in the present model was based on a precise reconstruction of a re- 
gion CA1 pyramidal neuron. The main additional assumptions involved the kinetics and 
distribution of the four membrane channels, and the dynamics of Ca2+in the neighborhood 
of influx. The forms assumed for these mechanisms were biophysically plausible, and the 
kinetic parameters were based on estimates from a collection of experimental studies (listed 
in Lytton and Sejnowski, 1991; Zador et al., 1990). Variation within the range of uncer- 
tainty of these parameters did not alter the main conclusions. The chief untested assump- 
tion of this model was the existence of cold spots. Although there is experimental evidence 
....... i ' ' ..... 
 ..."'.'.':'.oI ........ 
..........  i + + i 
Input. A -, 
Fig. 4 Nonlinear pattern separation Neuronal f'aing is represented as a joint function of two input 
pathways (A/B). Filled circles indicate that the neuron fired for the indicated stimulus parmeters. 
Some indication of the stochastic nature of this function, resulting form the noisy background, is giv- 
en by the density of interdigitation of points and circles. 
56 Zador, Claiborne, and Brown 
supporting the presence of both Ca and AHP channels in the dendrites, there is at present 
no direct evidence regarding their colocalization. 
4.2 COMPUTATIONS IN SINGLE NEURONS 
4.2.1 Neurons and Processing Elements 
The limitations of the McCulloch and Pitts (1943) PE as a neuron model have long been 
recognized. Their threshold PE, in which the output is the weighted sum of the inputs 
passed through a threshold, is static, deterministic and treats all inputs equivalently. This 
model ignores at least three key complexities of neurons: temporal, spatial and stochastic. 
In subsequent years, augmented models have attempted to capture aspects of these com- 
plexities. For example, the leaky integrator (Caianiello, 1961; Hopfield, 1984) incorpo- 
rates the temporal dynamics implied by the linear RC component of the circuit element 
pictured in Fig. lB. We have demonstrated that the input-output function of a realistic neu- 
ron model can have qualitatively different behavior from that of a single processing ele- 
ment (PE). 
4.2.2 Interactions Within The Dendritic Tree 
The early work of Rall (1964) stressed the spatial complexity of even linear dendritic mod- 
els. He noted that input from different synapses cannot be considered to arrive at a single 
point, the soma. Koch et al. (1982) extended this observation by exploring the nonlinear 
interactions between synaptic inputs to different regions of the dendritic tree. They empha- 
sized that these interactions can be local in the sense that they effect subpopulations of syn- 
apses and suggested that the entire dendritic tree can be considered in terms of electrically 
isolated subunits. They proposed a specific role for these subunits in computing a veto 
an analog AND-NOT that might underlie directional selectivity in retinal ganglion cells. 
The veto was achieved through inhibitory inputs. 
The underlying neuron models of Koch et al. (1982) and Rall (1964) were time-varying but 
linear, so it is not surprising that the resulting nonlinearities were monotonic. Much steeper 
nonlinearities were achieved by Shepherd and Brayton (1987) in a model that assumed ex- 
citable spines with fast Hodgkin-Huxley K and Na channels. These channels alone could 
implement the digital logic operations AND and OR. With the addition of extrinsic inhibi- 
tory inputs, they showed that a neuron could implement a full complement of digital logic 
operations, and concluded that a dendritic tree could in principle implement arbitrarily 
complex logic operations. 
The emphasis of the present model differs from that of both the purely linear and of the dig- 
ital approaches, although it shares their emphasis on the locality of dendritic computation. 
Because the cold spot involved strongly nonlinear channels, it implemented a nonmonoton- 
ic response function, in contrast to strictly linear dendritic models. At the same time, the 
present model retained the essentially analog nature of intraneuronal signalling, in contrast 
to the digital dendritic models. This analog mode seems better suited to processing large 
numbers of noisy inputs because it preserves the uncertainties rather than making an imme- 
diate decision. Focussing on the analog nature of the response eliminated the requirement 
for operating within the digital range of channel dynamics. 
The NMDA receptor-gated channel can give rise to an analog AND with a weaker voltage- 
dependence than that induced by fast Na and K channels. Mel (1992) described a model in 
which synapses mediating increases to both the NMDA and AMPA conductances were dis- 
tributed across the dendritic tree of a cortical neuron. When the synaptic activity was dis- 
Nonlinear Pattern Separation in Single Hippocampal Neurons 57 
tributed in appropriately sized clusters, the resulting neuronal response function was 
reminiscent of that of a sigma-pi unit With suitable preprocessing of inputs, the neuron 
could perform complex pattem discrimination. 
A unique feature of the present model is that functional inhibition arose from purely exci- 
tatory inputs. This mechanism underlying this inhibition rathe AHP current was intrinsic 
to the membrane. In both the Koch et al. (1982) and Brayton and Shepherd (1987) models, 
the veto or NOT operation was achieved through extrinsic synaptic inhibition. This requires 
additional neuronal circuitry. In the case of a dedicated sensory system like the direction- 
ally selective retinal granule cell, it is not unreasonable to imagine that the requisite neu- 
ronal circuitry is hardwired. In the limiting case of the digital model, the requisite circuitry 
would involve a separate inhibitory intemeuron for each NOT-gate. 
4.2.3 Adaptive Dendritic Computation 
What algorithms can harness the computational potential of the dendritic tree? Adaptive 
dendritic computation is a very new subject. Brown et al. (1991, 1992) developed a model 
in which synapses distributed across the dendritic tree showed interesting forms of spatial 
self-organization. Synaptic plasticity was governed by a local biophysically-motivated 
Hebb rule (Zador et al., 1990). When temporally correlated but spatially uncorrelated in- 
puts were presented to the neuron, spatial clusters of strengthened synapses emerged within 
the dendritic tree. The neuron converted a temporal correlation into a spatial correlation. 
The computational role of clusters of strengthened synapses within the dendritic tree be- 
comes important in the presence of nonlinear membrane. If the dendrites are purely pas- 
sive, then saturation ensures that the current injected per synapse actually decreases as the 
clustering increases. If purely regenerative nonlinearities are present (Brayton and Shep- 
herd, 1987; Mel, 1992), then the response increases. The cold spot extends the range of 
local dendritic computations. 
What might control the formation and distribution of the cold spot itself?. Cold spots might 
arise from the fortuitous colocalization of Ca and KAH P channels. Another possibility is 
that some specific biophysical mechanism creates cold spots in a use-dependent manner. 
Candidate mechanisms might involve local changes in second messengers such as [Ca 2+] 
or longitudinal potential gradients (cf. Poo, 1985). Bell (1992) has shown that this second 
mechanism can induce computationally interesting distributions of membrane channels. 
4.3 WHY STUDY SINGLE NEURONS? 
We have illustrated an important functional difference between a single neuron and a PE. 
A neuron with cold spots can perform extensive local processing in the dendritic tree, giv- 
ing rise to a complex mapping between input and output. A neuron may perhaps be likened 
to a "micronet" of simpler PEs, since any mapping can be approximated by a sufficiently 
complex network of sigmoidal units (Cybenko, 1989). This raises the objection that since 
micronets represent just a subset of all neural networks, there may be little to be gained by 
studying the properties of the special case of neurons. 
The intuitive justification for studying single neurons is that they represent a large but high- 
ly constrained subset that may have very special properties. Knowledge of the properties 
general to all sufficiently complex PE networks may provide little insight into the proper- 
ties specific to single neurons. These properties may have implications for the behavior of 
circuits of neurons. It is not unreasonable to suppose that adaptive mechanisms in biolog- 
ical circuits will utilize the specific strengths of single neurons. 
58 Zador, Claiborne, and Brown 
Acknowledgments 
We thank Michael Hines for providing NEURON-MODL assisting with new membrane 
mechanisms. This research was supported by grants from the Office of Naval Research, 
the Defense Advanced Research Projects Agency, and the Air Force Office of Scientific 
Research. 
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