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A Topology Inference Method of Cortical Neuron Networks Based on Network Tomography and the Internet of Bio-Nano Things


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This letter presents a topology inference technique for neuronal networks of the cortex of the human brain based on network tomography theory. We envision that this technique will be used for high-resolution and high-precision brain tissue tomography and imaging using principles of the Internet of Bio-Nano Things. Our network tomography solution relies on the classification of processed data of spike delay and synaptic weight functions of neuronal network activity. For a 6-layer cortical neural network, we achieved 99.27% of accuracy using the Decision Tree machine learning technique for individual neurons, 2-leaf and 4-leaf star topologies of neuronal networks.
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A Topology Inference Method of Cortical Neuron
Networks based on Network Tomography and the
Internet of Bio-Nano Things
Michael Taynnan Barros, Harun Siljak, Alaa Ekky and Nicola Marchetti
Abstract—This letter presents a topology inference technique
for neuronal networks of the cortex of the human brain based
on network tomography theory. We envision that this technique
will be used for high-resolution and high-precision brain tissue
tomography and imaging using principles of the Internet of Bio-
Nano Things. Our network tomography solution relies on the
classification of processed data of spike delay and synaptic weight
functions of neuronal network activity. For a 6-layer cortical
neural network, we achieved 99.27% of accuracy using the
Decision Tree machine learning technique for individual neurons,
2-leaf and 4-leaf star topologies of neuronal networks.
Index Terms—IoBNT, Molecular Communications, Network
Tomography, Topology Inference, Machine Learning
The tomography of the brain has historically provided med-
ical practitioners with an advanced technological apparatus
for observing and diagnosing the tissue state and its possible
pathologies. More recently, image processing techniques from
computerised tomography became an essential part of the
diagnosis of brain disorders. However, current computerised
tomography is still far from achieving high-resolution char-
acterisation of brain activity with network-level precision.
The most advanced computerised brain tomography, within
the minimally invasive space, is based on photoacoustics can
only reach precision at functional brain regions [1]. The
efficiency of diagnosis of brain disorders can be improved
with techniques that precisely map neuron connections at the
tissue level, as well as allow in-vivo characterisation of their
activity, based on advanced minimally invasive technologies.
The monitoring of in-vivo cellular activity will potentially
be achieved with implantable bio-nano devices for future
theranostics technology [2]. The idea behind this technology is
that cellular signals can be measured at the intracellular level,
intercellular level and/or extracellular level with biocompatible
bio-nano devices that interface with organs using connections
to individual cells. The newly proposed Internet of Bionano
Things (IoBNT) [3], envisions that these devices can have
added programmable functionalities and be remotely con-
trolled by developing technologies that allow their integration
with the Internet. The above-mentioned principles inspire
researchers to design minimally invasive solutions based on
bio-nano sensing that can assist in the future technologies for
M. T. Barros is with the TSSG/Waterford Institute of Technology, Ireland,
and CBIG at BioMediTech Unit, Faculty of Medicine and Health Technology
of the Tampere University, Finland. H. Siljak, A. Ekky and N. Marchetti are
with Trinity College Dublin, Ireland. The main contact is:
Spike Data
Processi ng
Learni ng
System Setup
Theoretical Neuron
Fig. 1: The envisioned network tomography system.
the brain, and in our case, we want to develop an IoBNT
approach for the next-generation tomography of brain tissues.
In IoBNT, bio-nano devices transmit information using
tissue as a communication channel through the molecular
communications paradigm [4], [5]. We envision that this
system will be used to create a distributed sensor system that
characterises the whole tissue using a limited number of bio-
nano sensors. This opens the possibility to create a technology
that sends back-back signals between bio-nano devices that
can be measured in different points of the tissue to infer
the actual network topology of cells. Such an approach is
called network tomography in computer networks [6], and is
depicted for our scenario in Fig. 1. Since the biological neuron
networks exhibit a great difference of transmission and signal
propagation depending on the type of cells [7], inferring the
organisation of these cells based on network topology models
is very challenging but also promises to achieve a precise
inference of the tissue state, or of the channel state.
To this end, we propose the usage of network tomography
for topology inference of neuronal cells in the brain towards
high-resolution characterisation of neuronal activity, or brain
neuron network mapping. We use the approach proposed by
Tsang et. al. that relies on estimation based on a delay measure
for topology inference [8] and extend it to include synaptic
weight, which is a natural neuronal molecular communication
process [9]. We use machine learning as our classification
method for different types of neuronal networks. We use the
NEURON simulator with models from the Blue Brain Project
to create a realistic representation of cortical microcircuit
networks. Then we apply a Decision Tree model for the
inference of three types of topologies, i.e., individual neurons,
2-leaf star and 4-leaf star. Our results show that 99.34% of
accuracy can be achieved with a delay of 5ms and synaptic
weight of 1.5, which are promising results that pave the
way for more research on this technology aiming to reliably
infer larger topologies structures and their eventual complex
communication patterns in the near future.
Inspired from the multi-compartment model of the Neocorti-
cal Microcircuit Collaboration Portal (NMC), here we present
the signal propagation model of a cortical neuron network that
is directly implemented in the NEURON simulator as well as
the network tomography theory with delay and synaptic weight
measures. In Cable theory, each neuron type can be simplified
to a sequence of capacitances and resistances in parallel and
then grouped to form an entire cortical microcircuit. The
Telegrapher’s Equation and the Compartment Models yields
to the following famous equation [10]
∂x2=VLV, (1)
in which τis the leakage conductance decay rate, Vis the
membrane voltage, xis the neurite1axis length, λis the
spatial coordinate decay rate and VLis the leakage (or resting)
potential of the cell. If we assume two neurons, labelled ηand
ω, the direction communication from ωto ηwith conductance
gηω , can be incorporated into Eq. (1) as suggested by the Rall
Model [11], and represented as follows:
VLVη+δ(xxg)gηω (VωVη) + X
in which δ(.)is the Dirac delta function, xgis the compartment
location, and Iion is the current from the n-th channel for
chemical synapses. The ionic channels depend on the specific
neuron chemical synapses in each cortical layer. Here they are
simplified to fit experimental data. We use the NEURON sim-
ulator that discretises both space and time, thus replacing the
partial differential equation by a coupled system of ordinary
differential equations (ODEs). This is called compartmental
modelling and has been introduced to neural modelling by
[11]. The spatial discretisation yields to a Kirchhoff’s current
law type equation described as:
Iionxg,n =X
in which cxgis the membrane capacitance of the compartment,
and Rthe resistance. The right-hand side of Eq. (3) is the
sum of axial currents that enter this compartment from its
adjacent neighbours, including neighbouring cells in case of
cellular border compartments. When Vxg> th, where th is
the excitation threshold, the neuron will produce a spike. Over
time, it will produce a set of spikes leading to a signal S(t),
which represents the spike train.
1projection from the cell body of a neuron, which can be either an axon
or a dendrite.
Typical network tomography can be reduced to a linear
model of a vector of measurements y, which represents a
set of all the calculated multi-domain measurements of the
network that is defined as y=A(θdθw) + , in which Ais
a connection probability matrix between cortical neurons, θd
is a vector of signal propagation delay, θwis a vector of signal
propagation synaptic weight, is a generic noise matrix with
same shape as Aand is the Hadamard product. Since we
consider small topologies, we can assume that all elements in
are 0. Now we use the following equations for calculating all
the delay measures for θdand the synaptic weight measures
for θw. Let us define dk=dtdt1,{dt, dt1} ∈ S(t),
in which dtis the time of the current spike, dt1is the
time of the previous spike, and the average delay will be
D= (1/K)(PK
k=0 dk), where Kis the total amount of delay
measures. The synaptic weight values Win θware considered
to be constant in the NEURON models. In this letter, we aim to
demonstrate the impact of machine learning-based inference of
network tomography as an advancement of the work by Tsang
et. al. [8], and therefore the communication system between
bio-nano machines is considered ideal and the study of its
scalability is left for future work. In the next sections, we will
build a Decision Tree Classification model based on the values
of yfor different topologies.
We aim to evaluate the impact of the vector of measure-
ments y, in particular variations of spike delay and synap-
tic weight, on the prediction of an individual neuron, a 2-
leaf star and 4-leaf star topologies. We use metrics such as
accuracy, precision and recall to measure the effectiveness
of the proposed inference technique. We define accuracy as
the percentage of correctly classified topologies, precision
represents the percentage of topologies that are properly linked
to the predicted class, and recall represents the percentage of
topologies that are correctly identified.
1) Simulation Setup: The cerebral cortex or neocortex con-
sists of six layers where the majority of neurons are arranged
vertically while the most abundant neurons are the efferent
pyramidal cells. We used experimental data to adjust Eq. (3) of
these cell types from the digital reconstruction of the microcir-
cuitry of somatosensory cortex2. Our models comprised of six
neurons, one from each layer with different morphology types
(m-types). For simplicity neurons were arbitrarily picked, but
the m-type was dependent on whether the pathways were
excitatory or inhibitory. This was important for the spike train
analysis as excitatory neurons generate a net positive voltage
that exceeds the threshold potential and causes an Action
Potential. Thus, the neuron with excitatory properties was the
one stimulated. One neuron from each layer was used in three
topologies. The individual cell used was from Layer 5. The
two-leaf topology consisted of neurons from Layers 1, 4 and 6
while the four-leaf topology consists of neurons from Layers
1, 2/3, 4, 5 and 6 respectively. Morphological and biophysical
details of the neurons were initialized before arranging them
2The experimental data of juvenile rat, the digital reconstruction and the
simulation results are available at the Neocortical Microcircuit Collaboration
Portal (NMC Portal;
a) b)
L6 L1
Fig. 2: The used topologies (a) Two Leaf Star (b) Four-Leaf Star
in a star topology on a unit circle for Eq. (3). The base of
the star was placed at the origin so L6 was placed at (0, 0, 0)
three dimensional Cartesian coordinates for both topologies.
This was enough for the individual neuron model. The two-
leaf topology is displayed in Fig. 2 (a), and is defined with
L1 placed to the right of L6 using the unit circle with co-
ordinates (2π, 0, 0). L4 was placed to the left of the origin
with co-ordinates (π, 0, 0). For the four-leaf topology, Fig. 2
(b), the two-leaf topologies configuration is used by two other
cells, and L2/3 and L5 models were added. L2/3 model was
placed below L6 with co-ordinates (0, 3π/2, 0) while L5 was
placed above L6 with co-ordinates (0, π/2, 0). The neuron
at the origin was stimulated in multiple points using virtual
synapses which will drive the cell through synaptic events
(located in orange in Fig. 2). This is important to generate
data for training the machine learning algorithm used for the
inference process. An axon from the cell at L6 is connected
to a synapse in the middle of the dendrite on the target
cell with a certain connection probability. The neuron models
were connected and stimulated with a simulation time of
1000ms. From the spike train signals we obtained the voltage,
time, delay and frequency of spikes, number of spikes and
average power spectral density of each spike were recorded
and considered as machine learning features. To access the
neuron experimental data in the NMC portal, the NEURON
simulator was used and RapidMiner, a machine learning tool,
was used for inferring cortical topologies. We used NEURON
to generate yvalues creating an overall system data set. Then
we created a Decision Tree Classification model implemented
in RapidMiner with a 30-70 split (30% testing and 70%
training) to classify the topologies investigated running a
cross-validation methodology. We chose Decision Trees-based
models due to low computing complexity in both training and
testing phases, which is more suitable to be implemented in
bio-nano devices. However, the performance of these models
might change, and should be investigated in future works.
2) Classification of Individual Neuron Models: A Layer
5 model was investigated for the individual neuron anal-
ysis, specifically the L5 DBC cAC model. A combi-
nation of 4 presynaptic m-types originating from lay-
ers L2/3 (MC SP STPC BP), L4 (PC NBC UTPC), L5
(PC LBC IPC) and L6 ( PC STPC IPC) respectively. The
spike trains for each simulation are shown, see Fig. 3. The
spike train in Fig. 3 tend to follow the same pattern at the
start of the simulation where the neuron cell remains at a
stable voltage. However, they deviate at the occurrence of the
first spike. The first spike occurs at 783ms in Fig. 3 (a), but
that is not the case for Fig. 3 (d) where it occurs at 765ms.
Another significant difference is the number of spikes present
Fig. 3: Spike Trains for Individual Neuron Model of L5 DBC cAC
in the spike train; Fig. 3 (d) contains 100 spikes while Fig.
3 (a) contains 71 spikes. A very interesting observation was
that a presynaptic m-type from a Pyramidal Cell (PC) had a
greater number of spikes than a presynaptic m-type from a
Martinotti Cell (MC) during this simulation. This shows that
when different m-types experience the same stimulation, they
react in different ways. Moreover, PC type neurons (Fig. 3
b,c and d) observed a smaller delay between spikes as well as
more frequent spikes when compared to MC neuron (Fig. 3
a). This validates our model based on the findings in [12]
where the pyramidal cell in Layer 2/3 experienced higher
frequency spikes when compared with other m-types in higher
layers. We analysed the performance of classifying individual
neuron models. The obtained accuracy results in predicting the
presynaptic m-types was 96.95%. Obtaining a high accuracy
result paves the way for investigating multiple neuron models
connected in a star topology.
3) Inferring Multiple Neuron Models in a Network: Here
we aim to focus our analysis on three different topologies that
were previously presented: Individual Neuron, 2-Leaf Star and
4-Leaf Star. We attempt to investigate the effect of the synaptic
weight (herein called weight) and delay only on the 2-Leaf and
4-Leaf topologies inference-based classification. We directly
obtain the inference results from the classification outputs for
analysis simplicity, however probabilistic approaches can be
used to increase the accuracy of inference results, for example,
using Maximum Likelihood methods. Fig. 4 a), b) and c) show
the results for the comparison of spike trains with weights (W)
1.5 and 3 combined with delays (D) 5ms and 10ms for the
2-Leaf (top) and 4-Leaf topologies (bottom). For W= 1.5
and D= 5ms 4 (see Fig. 4a), the cell located at the base of
the star in both topologies was stimulated for 1000ms and the
time and voltage were recorded. From 0 to 100ms, the cell
experiences multiple synaptic events with a delay of 5ms. The
4 star-leaf topology exhibits a continuous spike train response
compared with the 2-leaf topology. This behaviour results from
the composition (e-type) present in the cell, named continuous
Adapting (cAD), where the cell continuously adapts to changes
or synaptic events. For W= 1.5and D= 10ms (see Fig.
4b), the delay between synaptic events has increased resulting
in a decrease in the number of spikes over the simulation
run time. This delay change is independent of the topology
thus affecting all neurons. For W= 3 and D= 5ms (see
Fig. 4c), unlike doubling the delay, the result of doubling the
weight increases the number of spikes for the same simulation
run time. The voltage is not affected as increasing weight
only increases the number of spikes and frequency of spikes.
We designed a decision tree model using the RapidMiner
tool to determine topologies given a training phase with the
Fig. 4: Results from the topology inference technique a decision tree classification algorithm. a) show spike train for a 2-Leaf (top) and
a 4-Leaf (bottom) star topology with weight of 1.5 and delay of 5ms b) same scenario now with the weight 1.5 and delay 10ms c) same
scenario now with the weight 3 and delay 5ms d) accuracy results for all weights (W) and delays (D) e) precision and recall results for the
individual, 2-Leaf and 4-Leaf star topologies for W=1.5 and D=5ms (left), W=1.5 and D=10ms (center) and W=3 and D=10ms (left).
neuronal communication parameters as features, including the
delay, synaptic weight, and spike rate. Overall accuracy of the
classification was obtained for all analysed delay and weight
values (see Fig. 4d). An overall accuracy of 99.27% was
obtained for the classifier on the entire data set, with all the
weights and delay values for all topologies. Varying delay
and weight both affect the overall accuracy when classify-
ing topologies. Doubling the delay gives the lowest overall
accuracy at 99.11% as the number of spikes and time are
not enough for predicting topologies. However, doubling the
weight gives a higher accuracy at 99.37%. This accuracy was
obtained for small star topologies which contained a maximum
of 5 neuron models. Class precision and class recall were
used to evaluate the quality of prediction for each application
class (see Fig. 4e). The lowest value of class precision was
obtained when predicting the 2 Leaf topology as 63 of the
67 instances were correctly classified but the remaining 4
were misclassified as the 4 Leaf topology. Similarly, the class
recall was lowest for the 2 Leaf topology. An increase in
misclassification leads to a decrease in class recall for that
same topology. It is evident that spikes might occur at the
same times in both 2 Leaf and 4 Leaf topologies with the delay
increase, leading, thereafter, to a decrease in overall accuracy
and recall.
In this letter, we propose a network tomography tech-
nique based on delay and synaptic weight analysis to infer
cortical circuits of the brain towards the next-generation of
computerised imaging based on bio-nano sensing technology.
We analysed the performance of our approach based on the
Decision Tree technique. We obtained an overall accuracy of
99.27% including all dataset for varying spike transmission
delay and synaptic weight. In reality, the cortical circuit
contains hundreds of thousands of connected neuron models
which is very challenging to infer with the current training
model we have. However, our results show that there is a lot
of potential for this technology since small topologies can be
inferred with great accuracy, precision and recall. Now the
challenge lies in hugely scaling this technique for hundreds
and then hundreds of thousands of neurons.
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Neuroscience is at a crossroads. Great effort is being invested into deciphering specific neural interactions and circuits. At the same time, there exist few general theories or principles that explain brain function. We attribute this disparity, in part, to limitations in current methodologies. Traditional neurophysiological approaches record the activities of one neuron or a few neurons at a time. Neurochemical approaches focus on single neurotransmitters. Yet, there is an increasing realization that neural circuits operate at emergent levels, where the interactions between hundreds or thousands of neurons, utilizing multiple chemical transmitters, generate functional states. Brains function at the nanoscale, so tools to study brains must ultimately operate at this scale, as well. Nanoscience and nanotechnology are poised to provide a rich toolkit of novel methods to explore brain function by enabling simultaneous measurement and manipulation of activity of thousands or even millions of neurons. We and others refer to this goal as the Brain Activity Mapping Project. In this Nano Focus, we discuss how recent developments in nanoscale analysis tools and in the design and synthesis of nanomaterials have generated optical, electrical, and chemical methods that can readily be adapted for use in neuroscience. These approaches represent exciting areas of technical development and research. Moreover, unique opportunities exist for nanoscientists, nanotechnologists, and other physical scientists and engineers to contribute to tackling the challenging problems involved in understanding the fundamentals of brain function.
Conference Paper
The high level of connections and hierarchy in the cortical microcircuits communication channel characterises a complex system that yet neuroscientists cannot understand it fully. The quantification of the communication channel capacity of cortical microcircuits enables a further understanding of information transfer inside that microscale of the brain. In this paper, we present a cortical microcircuit communication channel capacity using a novel integrated approach to cortical microcircuit data and information theory. We use data from the Digital Reconstruction of Neocortical Microcir-cuitry portal that accounts for detailed neurological characteristics including neuron morphology, connection probabilities, neuronal noise, and cortex connection hierarchy. Our results show a constant significant decrease of system's capacity over time, noise and neuronal stimulation.We also observed an independent hierarchical property between arbitrary layers, meaning that the configuration of the cortical microcircuit is crucial to define its communication performance. Based on these observations, we find the first piece of evidence that the cortical microcircuit channel behaviour resembles a typical fast fading channel.
Understanding the communication theoretical capabilities of information transmission among neurons, known as neuro-spike communication, is a significant step in developing bio-inspired solutions for nanonetworking. In this study, we focus on a part of this communication known as synaptic transmission for pyramidal neurons in Cornu Ammonis (CA) area of hippocampus location in the brain and propose a communication-based model for it that includes effects of spike shape variation on neural calcium signaling and the vesicle release process downstream of it. For this aim, we find impacts of spike shape variation on opening of voltage-dependent calcium channels (VDCCs), which control the release of vesicles from pre-synaptic neuron by changing the influx of calcium ions. Moreover, we derive the structure of optimum receiver based on Neyman-Pearson detection method to find the effects of spike shape variations on the functionality of neuro-spike communication. Numerical results depict that changes in both spike width and amplitude affect the error detection probability. Moreover, these two factors do not control the performance of the system independently. Hence, a proper model for neuro-spike communication should contain effects of spike shape variations during axonal transmission on both synaptic propagation and spike generation mechanisms to enable us to accurately explain the performance of this communication paradigm.
Photoacoustic computed tomography (PACT) is a non-invasive imaging technique offering high contrast, high resolution, and deep penetration in biological tissues. We report a PACT system equipped with a high frequency linear transducer array for mapping the microvascular network of a whole mouse brain with the skull intact and studying its hemodynamic activities. The linear array was scanned in the coronal plane to collect data from different angles, and full-view images were synthesized from the limited-view images in which vessels were only partially revealed. We investigated spontaneous neural activities in the deep brain by monitoring the concentration of hemoglobin in the blood vessels and observed strong interhemispherical correlations between several chosen functional regions, both in the cortical layer and in the deep regions. We also studied neural activities during an epileptic seizure and observed the epileptic wave spreading around the injection site and the wave propagating in the opposite hemisphere.
Nanomedicine is revolutionizing current methods for diagnosing, treatment and prevention of diseases with the integration of molecular biology, biotechnology as well as nanotechnology for sensing and actuation capabilities at the molecular scale using nanoscale devices, namely nanomachines. While numerous examples of these applications have been tested in vivo, the real deployments are far from reality. Limitations in controlling, monitoring, miniaturization, and computing inhibit access and manipulation of information at the nano-scale. Integrating communication and networking functionalities provide new opportunities for such challenges with the newly introduced Molecular Communications. These natural communication systems are found with plurality inside the human body. The current challenge is to utilize these natural systems to create artificial biocompatible communication networks that can interconnect multiple nanomachines in multiple parts of the body and connected to the cloud, is defined as the Internet of Bio-Nano Things (IoBNT). Nanonetworks inside cellular tissues perform communication using a signaling process such as Ca²⁺. This specifically signaling process is very important for many regulatory functions in tissues and its control and communication is crucial to allow nanomedicine capabilities towards diagnosis and treatments of diseases at the nano-scale. This paper presents a review of techniques that enable the design of the Ca²⁺-signaling-based molecular communication system for cellular tissues, essential tools for its deployment, application and, lastly, the research future direction in this field. In the end, one must acquire the sufficient knowledge to understand both biological and telecommunication concepts that encompass this technology to bring it further.
Molecular communication (MC) is the most promising communication paradigm for nanonetwork realization since it is a natural phenomenon observed among living entities with nanoscale components. Since MC significantly differs from classical communication systems, it mandates reinvestigation of information and communication theoretical fundamentals. The closest examples of MC architectures are present inside our own body. Therefore, in this paper, we investigate the existing literature on intrabody nanonetworks and different MC paradigms to establish and introduce the fundamentals of molecular information and communication science. We highlight future research directions and open issues that need to be addressed for revealing the fundamental limits of this science. Although the scope of this development encompasses wide range of applications, we particularly emphasize its significance for life sciences by introducing potential diagnosis and treatment techniques for diseases caused by dysfunction of intrabody nanonetworks.
We present a first-draft digital reconstruction of the microcircuitry of somatosensory cortex of juvenile rat. The reconstruction uses cellular and synaptic organizing principles to algorithmically reconstruct detailed anatomy and physiology from sparse experimental data. An objective anatomical method defines a neocortical volume of 0.29 ± 0.01 mm(3) containing ∼31,000 neurons, and patch-clamp studies identify 55 layer-specific morphological and 207 morpho-electrical neuron subtypes. When digitally reconstructed neurons are positioned in the volume and synapse formation is restricted to biological bouton densities and numbers of synapses per connection, their overlapping arbors form ∼8 million connections with ∼37 million synapses. Simulations reproduce an array of in vitro and in vivo experiments without parameter tuning. Additionally, we find a spectrum of network states with a sharp transition from synchronous to asynchronous activity, modulated by physiological mechanisms. The spectrum of network states, dynamically reconfigured around this transition, supports diverse information processing strategies. Paperclip: VIDEO ABSTRACT.
This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre (Hodgkinet al., 1952,J. Physiol.116, 424–448; Hodgkin and Huxley, 1952,J. Physiol.116, 449–566). Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical form (Section 2) and to show that they will account for conduction and excitation in quantitative terms (Sections 3–6).
This paper is concerned with the interpretation of passive membrane potential transients produced in a neuron when intracellular microelectrodes are used to apply current across the soma membrane. It is also concerned with the specific problem of estimating the nerve membrane time constant from experimental transients in neurons having extensive dendritic trees. When this theory is applied to the most recent results published for cat motoneurons, the resulting membrane time constant estimates are significantly larger than the values estimated by Eccles and collaborators. The time course of soma membrane potential is solved for a variety of applied currents: current step, brief pulse of current, sinusoidal current, voltage clamping current, and a current of arbitrary time course. The sinusoidal case provides a theoretical basis for a purely electrophysiological method of estimating the fundamental ratio between combined dendritic input conductance and soma membrane conductance. Also included is the time course of passive decay to be expected to follow various somadendritic distributions of membrane depolarization or hyperpolarization. The discussion includes an assessment of the observations, hypotheses, and interpretations that have recently complicated our understanding of synaptic potentials in cat motoneurons. It appears that electrotonic spread between the dendrites and soma can account for the observations which led Eccles and collaborators to postulate a prolonged residual phase of synaptic current in cat motoneurons.