Certain studies have showed that synapses have a stable
influence that is independent of their distance to the soma, a
phenomenon referred to as “equal vote” or “dendritic
democracy” , . In this setting, the synaptic weights
and time constant would be modified in order to obtain the
same EPSP at the soma level for all the synapses i.e.
distance-dependent scaling . If we consider that the
propagation times remain different, we can calculate a
composite EPSP with stable amplitude of several tens of
milliseconds. To obtain this type of effect in the soma it is
necessary to organize the synapses in a regular manner along
the dendrite with the synaptic weights increasing with
distance to the soma. We will call this particular
configuration the “frequency sensitive setting” and all the
others “spatio-temporal sensitive”. Here we demonstrate that
neurons organized in a frequency sensitive manner (hereafter
named frequential neurons) are able to « count » the spikes
that arrive within a specific time interval, a phenomenon
which itself depends on the dendritic length and the
propagation speed. With a constant EPSP shape, the neuron
becomes particularly sensitive to the average firing rate of
the global pattern but reacts little to changes in the precise
timing of each spike
The term Single Fiber EPSP or composite EPSP was first
introduced by Burke , and later used by Sheppard 
page 91 and calculated by Larkum et al . The single fiber
EPSP is the result in the soma of the simultaneous activation
of a pool of synapses located at different positions on the
same dendritic branch. This co-activation is deterministic as
long as they are all synapses from the same pre-synaptic
neuron. In this setting, a single spike in a presynaptic cell
activates its entire pool of synapses. The generated EPSPs
are combined to create a composite EPSP in a deterministic
fashion, with S representing the number of synapses in the
pool and F the total number of incoming spikes arriving at tf
times. Using the previous equations 1 to 4 we can write the
equation of the composite EPSP, for each time t and for each
position p on the dendrite:
Location of a synapse on a dendrite will modulate the
arrival time of the EPSP at the soma level due to the
propagation time of the EPSP in the dendrite  and will
also modulate the shape of the alpha function. Fig 1B.
The soma arrival time depends logically on the
propagation velocity v within the dendrite, on the spike
arrival time at the synapse and on the distance to be
traversed by the EPSP to the soma: lambda_s, or synapse position.
Accounting for this dendritic effect, equation 1 then
becomes u(t ,p,lambda_s) with p relating to the position on the
dendrite relative to the soma:
μ represents the maximum value of u expressed without
considering the synaptic weight w. It is only used to
normalize u, making u equal the synaptic weight w at the
EPSP peak. μ is expressed by the following equation:
The amplitude of the EPSP is also attenuated during its
somatopetal propagation (from the synapse to the soma) ,
, . This attenuation at a particular position p on the
dendrite depends on the distance to the synapse:( ) and
on a chosen attenuation rate α. We used the following
What is the relation between the morphology of a neuron and its function? Neurons mostly present large dendritic extensions, the exact role of which largely remains unknown. It could be, as has already been suggested, that dendrites are simply there to increase the membrane surface area of the neurons to enable the binding of 10- or 20 fold more synapses (Mel WB in  p421). This being the case, the shape of the dendritic tree and the particular position of the synapses would have only little influence on the computation performed by the neurons. The best model for approximating this dendritic function would be the point neuron. A: the arrival of an action potential at a synapse triggers an EPSP in the post-synaptic neuron.  Pages 75-76.
This function supposes an incoming firing time at t=0 of the presynaptic neuron. In this equation, u is the membrane potential of the dendrite and soma, w is the synaptic weight, delta_ax is the transmission delay between the time the pre-synaptic neuron is active and the time the synapse s becomes active. t is the time constant, t_r is the rising time constant and H is the Heaviside function.
One of the most obvious functions that the point neuron model can perform is coincidence detection: when two neurons are synchronized and both have the same point neuron as output, then the simultaneous arrival of the spikes triggers a summed EPSP that overshoots the spike threshold. This target point neuron then emits an action potential that indicates its coincidence detection. Thus, for a point neuron, the variation of the membrane potential depends exclusively on the synaptic weights and the arrival times of the incoming spikes.
What is less clear however is how this type of neuron, except when integrated in large networks, could detect many spike arrival times, i.e. a specific spike train. To investigate this, we modified the point neuron equation to incorporate a
propagation function along a dendrite.
ULTIMATELY understanding the role of dendrites in neural computation requires a theory. This theory must identify the benefits of having dendrites and reveal the basic principles used to provide these benefits” . In the conclusion of the book entitled “Dendrites”, the editors Nelson Spruston, Greg Stuart and Michael Häusser wrote:
“Despite this tremendous progress, the most exciting times in dendrite research lie ahead of us. Much of the knowledge we have accumulated to date has been descriptive […] Two key questions that now need addressing are:
- What computations does each neuron perform within its neuronal network?
- And, which features of dendrites are most relevant to how neurons perform these computations? 
Concerning neuronal functions, a first element was given by Rieke et al in 1999 :
“When we see, we are not interpreting the pattern of light intensity that falls on our retina; we are interpreting the pattern of spikes that the million cells of our optic nerve send to the brain. […] Spike sequences are the language for which the brain is listening, the language the brain uses for its internal musings, and the language it speaks as it talks to the outside world.”
There is also much evidence showing that biological neurons are able to interpret firing rate  and temporal codes . Here we show that the addition of a single dendrite to a point neuron model considerably extends its possible range of functions, in particular, enabling it to detect a precise spatio-temporal pattern of spikes or, conversely, to have an undifferentiated response to precise spike timing but to react to average firing rate. We also show that these two distinct types of behavior characterized either by spatio-temporal or frequency sensitivity depend solely on the morphology of the « synapse-dendrite » system and in particular on the position of synapses on the dendrite. These findings provide an additional element necessary to our global understanding of the features required by dendrites that enable biological neurons to perform these computations.
How could synapse number and position on a dendrite affect neuronal behavior with respect to the decoding of firing rate and temporal pattern? I developed a model of a neuron with a passive dendrite and found that dendritic length and the particular synapse positions directly determine the behavior of the neuron in response to patterns of received inputs. I revealed two distinct types of behavior by simply modifying the position and the number of synapses on the dendrite. In one setting – spatio-temporally sensitive – the neuron responds to a precise spatio-temporal pattern of spikes, but shows little change following an increase in the average frequency of the same input pattern. In the other setting – frequency sensitive – the neuron is insensitive to the precise arrival time of each spike but responds to changes in the average firing rate. This would allow neurons to detect different spatio-temporal patterns.