# The Koha Code
A Biological Theory of Memory

Lum Ramabaja

###### Abstract

This work introduces the Koha model, a new theory that aims to explain two
unresolved phenomena within biological neural networks: How information is
processed and stored within neural circuits, and how neurons learn to become
pattern detectors. In the Koha model, the dendritic spines of a neuron serve
as computational units that scan for precise spike patterns in their synaptic
inputs. The model proposes the existence of a temporal code within each
dendritic spine, which is used for the dampening or amplification of signals,
depending on the temporal information of incoming spike trains. Compelling
evidence is provided and a concrete process is described for how signal
filtration occurs within spine necks. A competitive learning algorithm is then
proposed that describes how neurons use their internal temporal codes to
become pattern detectors.

_K_ eywords Memory $\cdot$ Dendritic Spines $\cdot$ neural code $\cdot$
competitive learning $\cdot$ associative memory

##  1 Introduction

###  1.1 Form follows function

The ability to form and store memories, recall past events, make inferences
and deductions, are all examples of "emergent algorithms" that arise from
neural circuits. Emergence occurs when a system exhibits characteristics that
its constituents do not possess but emerge because of rules, or interactions
between its parts. Parts of an emergent system often adhere to very basic
principles. When those parts interact with one another, complexity can arise.
In the case of the brain, neural circuits serve as the physical substrate for
emergent algorithms, which arise as a result of neuronal interactions. Any
modification to a neural circuit or structure may thus affect an existing
algorithm or, in certain instances, result in the creation of a new algorithm.
If a useful algorithm forms as a result of a mutation in one of its underlying
neural circuits, the organism with the altered algorithm will have a
competitive advantage over other organisms. Over time, evolutionary forces
will shape the underlying circuits in a way that makes the emergent algorithm
more efficient. This means that by examining the optimized "hardware" of
organisms, it might be possible to deduce some of the existing emergent
algorithms in nature. If a certain pattern repeats across many species, we can
be very certain that the repeating pattern plays a significant role. By
looking at the optimised morphology, structures, and processes of various
neurons, we can start to guess their purpose in the context of the mind. After
all, in biology, form follows function.

###  1.2 Patterns in the substrate

One of the most notable exitatory neurons is the pyramidal neuron. It is the
most common type of excitatory neuron in the mammalian cortical structures, as
well as one among the largest neurons in the brain. Each mammal, as well as
birds, fish, and reptiles, has pyramidal neurons. Apart from its wide
occurrence in nature, the pyramidal neuron exhibits also many noteworthy
physical features, particularly when contrasted to inhibitory interneurons.
The following is a list of key observations that are relevant for the Koha
model presented in section 2:

  * •

Studies in the layer 2/3 of the rat neocortex have shown that pyramidal
neurons posses about 10 times more local connections with fast-spiking
inhibitory neurons, than with other pyramidal neurons, with most of the
connections being reciprocal [1].

  * •

Pyramidal neurons have two kinds of dendrites. Basal dendrites, which come out
of the soma and branch extensively. And a single, long, dendrite that emerges
from its soma, also known as the apical dendrite. It can extend for several
hundred microns before branching into many smaller dendrites (see figure 1)
also known as the apical tuft. Basal dendrites tend to receive their input
from nearby neurons, whereas the apical dendrite receives its inputs from more
distant regions [2]. While pyramidal neurons have an apical dendrite,
inhibitory interneurons do not.  Figure 1: The variable structure of pyramidal
neurons across different parts of the brain. Even though the pyramidal neuron
differs slightly across layers, it is characterized by its stereotypical
morphology - the presence of an apical dendrite and the presence of basal
dendrites. Adapted from Spruston, 2008 [3] .

  * •

The apical dendrite and basal dendrites of pyramidal neurons are completely
covered with tiny protrusions structures known as dendritic spines (see figure
2). It is believed that dendritic spines serve as memory storage components
for neurons [4]. In contrast to excitatory neurons, most interneurons do not
posses, or posses very few dendritic spines, i.e. most interneurons are
essentially spineless [4].  Figure 2: Two dendrites covered with dendritic
spines. Adapted from Spruston, 2008 [3] .

  * •

There are various stages of dendritic spines, the exact borders of which are a
bit blurry. Traditionally spines are classified into thin, stubby, and
mushroom spines [5] (see figure 3). Thin spines have a small head and a long
neck that connects to the parental dendrite. Stubby spines have no neck at all
and are more expressed in infants [6]. Whereas mushroom spines have a wide
neck with a large head and are more expressed in adult brains [6, 7]. Because
of the properties each spine category has, it is believed that thin spines may
play a role in "learning", whereas mushroom spines in long-term memory [8].
Thin spines can dynamically appear and disappear throughout life, while most
mushroom spines remain stable for a lifetime [8, 9]. The stability of the
mushroom spine and its high expression in adult brains, make it an ideal
candidate as a long term "memory unit".  Figure 3: The three traditional types
of dendritic spines.

  * •

We now know that dendritic spines receive most of the excitatory inputs in
neurons, and that practically every dendritic spine has an excitatory synapse
on its head [10, 11, 12]. In other words, excitatory axons prefer for some
reason to terminate particularly on spines, they almost never connect directly
to dendritic shafts (the body of the dendrite) [12]. We can see the opposite
behavior in inhibitory axons. Inhibitory axons almost always connect directly
to dendritic shafts, rather than on spines [13]. The consistency of this
pattern suggests a functional value.

  * •

Imaging experiments have demonstrated that dendritic spines behave indeed like
electrical and biochemical compartments [14, 15, 16, 17, 18, 14]. The narrow
neck of the spine can create an isolated compartment in which the biochemical
signals of the spine head do not spread along the parent dendrite. Changes in
the length and narrowness of the spine neck, has a direct effect on the
compartmentalization of the spine [19].

  * •

Experiments also show that the most significant filtering of local potentials
in a neuron does not occur along the dendrite, but instead at the spine neck
[20]. The spine neck is somehow able to modulate the amplitude of incoming
signals. Further studies have also shown that spines, additionally to
filtering out local potentials, are also able to amplify them [21].

  * •

Evidence is now overwhelming that dendritic spines can change their physical
structure in seconds, following an appropriate stimulus [22, 23, 24, 25, 26].
Spine necks have an actin cytoskeleton (see figure 4) that can dynamically
change shape depending on its interactions with several actin regulators. This
change in geometry has a direct affect on local voltage amplification and
biochemical compartmentalization [27]. The strength of incoming signals can
therefore be dampened, or amplified, depending on the length and width that
the spine neck takes [28].  Figure 4: A simplified model of the organization
of actin in dendritic spines. Reprinted from [29] .

  * •

Some dendritic spines also posses an enigmatic organelle known as the spine
apparatus, a specialized formation of stacked endoplasmic reticulum that
occupies a large portion of the spine neck. Not all spines have a spine
apparatus, but almost all mushroom spines have one [30]. We now know that the
role of the spine apparatus is closely linked with local calcium trafficking
[31, 32, 33]. Due to the large surface area of the organelle, it is believed
that the spine apparatus may act as a calcium buffer [31]. Studies now show
that the spine apparatus is also able to quickly release $Ca^{2+}$ [34, 35].
The spine apparatus is thus acting as an intracellular calcium store [31],
serving as both calcium source and calcium sink, depending on internal
feedback.

  * •

The spine apparatus also appears to be closely associated with synaptopodin
[36], which is an Actin-associated Protein. In an experiment, synaptopodin-
deficient mice were associated with a loss of the spine apparatus and showed a
reduction in hippocampal long-term potentiation, suggesting that the spine
apparatus may play an important role in synaptic plasticity [37].
Synaptopodin’s preferential location in spines, and its close association with
the spine apparatus and the actin cytoskeleton, suggests that synaptopodin is
somehow involved in the rapid plasticity of the spine’s actin cytoskeleton. In
fact, one study suggested that synaptopodin may act as an actin-bundling
molecule in the spine neck, and that it may influence calcium release from the
spine apparatus [38].

  * •

An even more engimatic organelle is the cisternal organelle, found at the axon
initial segment of pyramidal neurons. The cisternal organelle is also a
formation of stacked endoplasmic reticulum and shares many structural
similarities with the spine apparatus. Very little is known about this
organelle. There is now evidence that it may serve as an intracellular calcium
store [39]. The cysternal organelle, just as the spine apparatus, also appears
to be closely associated with synaptopodin. Experiments showed that
synaptopodin-deficient mice were also associated with a loss of the cisternal
organelle at the axon initial segment [40].

###  1.3 Receptive fields and invariances

The portion of sensory space that can cause a neuron to fire when stimulated
is known as the neuron’s receptive field. There are visual receptive fields,
auditory receptive fields, olfactory receptive fields, and somatosensory
receptive fields. In the 1950s and 1960s, Hubel and Wiesel demonstrated that
different visual stimuli have different effects on the firing pattern of a
neuron [41, 42]. Hubel and Wiesel categorized the visual neurons in their
experiments into two groups: simple cells and complex cells.

Simple cells are neurons that fire whenever stimulated with a specific input
pattern. In the case of visual neurons, simple cells specialize to activate to
specifically oriented edges, or bars. These neurons are basic pattern
detectors that activate only to one specific input (see figure 5). In other
words, If the neuron "sees" its pattern, it will fire rapidly. If however the
pattern does not fully align with its receptive field, the neuron fires less
frequently, or not at all. The same principle of basic pattern detection
functions for other types of receptive fields as well. Simple cells with
auditory receptive fields for example respond selectively to sound
frequencies.

Figure 5: The response of a simple cell with a visual receptive field, when
stimulated with four different patterns. The first example shows a series of
action potentials, also known as spike train, when stimulated with an input
pattern with the optimum size, position, and orientation. The second example
shows a response to an input pattern that has an optimal orientation and size,
but a slightly unaligned position with the receptive field. The neuron emits
fewer action potentials. In the third example, the orientation of the input
pattern does not align at all with the receptive field of the neuron, causing
the neuron not to respond. In the last example, the input pattern represents a
large fully illuminated region, which does not align with the neuron’s
receptive field, and results in no firing.

Complex cells on the other hand, as the name suggests, have more complex
receptive fields. In the case of visual receptive fields, some complex cells
fire whenever stimulated with a specific input pattern, regardless of its
position in the receptive field (see figure 6 A). In other words, similarly to
artificial convolutional neural networks (CNNs) [43, 44], the receptive field
of these cells is said to have transition invariance. Other complex cells
might have a receptive field with a directional preference (see figure 6 C).
These cells fire whenever stimulated with an input pattern that moves at a
specific direction regardless of time, but do not fire if the same input
pattern moves across the receptive field in the opposite direction. Similarly
to artificial recurrent neural networks (RNNs) [45, 46], the receptive field
of these cells are said to have temporal invariance. Some complex cells fire
when stimulated with a specific pattern, regardless of the pattern’s size
inside the receptive field (see figure 6 D). These complex cells have a
receptive field that is scale invariant.

Figure 6: A \- The response of a complex cell with a visual receptive field,
when stimulated with four different patterns. The receptive field of this
complex cell is transition invariant. B \- The response of a complex cell with
a visual receptive field, when stimulated with four different Simpsons
snapshots. The receptive field of this complex cell fires whenever Bart is
inside the receptive field, but does not fire when Homer is presented. The
receptive field of this complex cell has elements of viewpoint invariance. C
\- A visual receptive field with a directional preference. The neuron fires
rapidly whenever its pattern moves across one direction, while remaining
inactive if the same pattern moves in the other direction. D \- A scale
invariant visual receptive field. The neuron fires when a specific input
pattern is used, regardless of the pattern’s scale.

With every successive processing stage, the receptive field of a neuron grows
in size and complexity. In the early processing stages, the receptive field of
neurons is specialized towards very specific input patterns, as seen in simple
cells. The receptive field of neurons in "deeper" processing stages start to
form various invariant abilities, such as transition invariance, scale
invariance, rotational invariance, etc., as seen in complex cells with visual
receptive fields. At a given stage, the receptive field of complex cells can
reach even more advanced abilities, such as recognizing objects regardless of
viewing angle (see figure 6 B). The receptive field of these complex cells are
said to have viewpoint invariance. How neurons form their given receptive
field and the various kinds of invariances remains unclear. It seems as if the
architecture of biological neural networks already contains the necessary
inductive biases in its design, to enable the learning of generalized
invariances.

###  1.4 Lateral inhibition

Besides excitatory pyramidal neurons, the brain also contains a wide spectrum
of specialized inhibitory interneurons. While pyramidal neurons are quite
large, have an apical dendrite, and can form connections with distant regions
of the brain, interneurons are smaller in size, do not have an apical
dendrite, and usually form many local connections.

Neural circuits containing both pyramidal neurons and interneurons can
organize in several ways, giving rise to networks with complex properties (see
figure 7). In this work, we are interested in the organization that enables
lateral inhibition. Lateral inhibition occurs when a pyramidal neuron
activates an inhibitory interneuron, which in turn suppresses the activity of
surrounding pyramidal neurons. Networks with lateral inhibitory configurations
(from now on simply referred to as "competitive circuits") form a "Winner take
all" competition between the pyramidal neurons within an area. The neuron that
fires before the other neurons "wins" the competition, by inhibiting the rest.

Figure 7: The various network configurations of inhibition. Circles represent
interneurons, whereas triangles represent pyramidal neurons. Upward arrows
indicate the activation of a neuron, whereas downward arrows indicate the
inhibition of a neuron. A sequence of vertical lines represents a spike train.
In a feedback network, the activation of a pyramidal neuron $A$, results in
the activation of an interneuron, which results in the inhibition of $A$. This
network organization might serve as a regulatory mechanism for a neuron’s
firing pattern. In a feed-forward network, the activation of an interneuron
results in the inhibition of another pyramidal neuron. A lateral inhibitory
configuration occurs when several pyramidal neurons share one ore more common
interneurons. In this network configuration, pyramidal neurons compete with
one another by trying to fire before the other pyramidal neurons, thereby
activating their common interneurons and inhibiting the rest. Illustration
reprinted from [47]

###  1.5 Temporal codes

"Individual nerve cells were formerly thought to be unreliable, idiosyncratic,
and incapable of performing complex tasks without acting in concert and thus
overcoming their individual errors. This was quite wrong, and we now realise
their apparently erratic behaviour was caused by our ignorance, not the
neuron’s incompetence." [48]

Neurons transmit and receive information via sequences of action potentials,
also known as spike trains. We now know that the number of action potentials
within spike trains is not the only variable used to encode information; the
timing of the emitted spikes can also be part of the code [49, 50, 51, 52,
53]. Recent evidence supports the existence of precise spike patterns with
millisecond-level precision (see figure 8) [54, 55, 56, 57, 58]. These
patterns have been found in the olfactory system [59, 50], auditory system
[60, 50], visual system [61, 52], somatosensory system [62, 63], and within
other sensory systems [64].

Figure 8: The detection of spatiotemporal spike patterns. The Y-axis of the
plot represents 100 neurons, 50 of which take part in a repeating precise
spike pattern. The X-axis represents time. The dots within the plot are
spikes. The darker dots represent spikes that are part of a precise spike
pattern. The bottom panel plots the overall spike counts over 10 ms time
intervals. The right panel plots the spike counts of each neuron over the
whole period. Reprinted from [65]. Modified from [66]

The role and mechanism behind the precise spike patterns remains however
enigmatic. Figure 8 perfectly demonstrates the elusiveness of the problem:
While looking at each neuron’s spike train, the spikes within the spike trains
seem completely random. At first glance, it looks as if there’s no temporal
information encoded within the spike trains. Only after performing an
exhaustive search, do the repeating millisecond-level precise spike patterns
become apparent (represented as black dots).

##  2 The Koha Model

The previous section addressed some of the biological observations required
for the Koha Model. This section on the other hand is speculative in nature.
In this section, I present a model which aims to explain two phenomena:

  1. 1.

How information is processed and stored within neural circuits.

  2. 2.

How neurons learn to become pattern detectors in an unsupervised way.

###  2.1 The Koha code

We have seen in section 1.2 that dendritic spines serve as electrical and
biochemical compartments, able to amplify, as well as dampen incoming signals
[21, 28]. In fact we now know that most of the filtering of local potentials
in a neuron happens within the spine neck and not along the dendrite [20]. The
spine is somehow able to know when to let information pass through its neck
and when not. The amplification, or dampening of incoming signals inside
dendritic spines directly depends on the length and narrowness of the spine
neck. Changes in the length and narrowness of the spine neck therefore, have a
direct effect on the compartmentalization of the spine [19]. We also know that
due to the spine neck’s highly dynamic actin cytoskeleton, dendritic spines
can change their physical structure in seconds, following an appropriate
stimulus [22, 23, 24, 25, 26]. This change in shape is governed by actin-
associated proteins such as drebrin [67], $\alpha$-actin [68], gelsolin [69],
fodrin [70], etc. The properties of these proteins in turn depend on the
calcium levels within a spine [38]. Increases in Intracellular calcium levels
might therefore change the properties of microfilament-associated proteins,
which then interact with the actin cytoskeleton of the spine, which results in
the reshaping of the spine neck, which in turn influences the
compartmentalization of the spine.

In short, if the dendritic spine could control its internal calcium levels in
response to incoming signals, then it could also dampen, or amplify those
signals by changing its shape. This is the main idea of the Koha model. Almost
all mushroom spines possesses an enigmatic organelle known as a spine
apparatus [30]. Spine apparati are now known to be intracellular calcium
stores that are able to quickly absorb, as well as release $Ca^{2+}$,
depending on internal feedback [31, 34, 35]. A fascinating study showed that
synaptic stimulation can cause the spine apparatus to release its internal
calcium stores [71]. According to the proposed hypothesis, I argue that there
is a molecular unit within the dendritic spine that encodes a temporal pattern
in the form of a sequence of "on" states and "off" states. Whenever a
dendritic spine receives a sequence of inputs, it will compare the spike
train’s temporal pattern, with its internal temporal code. If the received
spike train’s temporal pattern is similar to the spine’s temporal code,
information flows from the spine to its dendrite freely. If on the other hand
the spike train’s temporal pattern is significantly dissimilar from the spines
internal code, the spine apparatus gets notified to increase intracellular
calcium levels, which changes the shape of the spine and dampens the incoming
signals. Signal processing on a local level is therefore controlled
mechanically at the spine neck, the behavior of which depends on the temporal
pattern of the incoming spike train.

Figure 9: A \- The response of a dendritic spine when stimulated with a spike
train whose temporal code is similar to that of the spine’s internal code. The
spine neck remains unchanged and the signal can pass freely. B \- The response
of a dendritic spine when stimulated with a spike train whose temporal code is
not similar to that of the spine’s internal code. The spine neck becomes
longer and narrower, dampening the strength of the incoming signal in the
process.

###  2.2 Specialization through competition

To summarize, every dendritic spine on the apical dendride of a pyramidal
neuron computes a distance measure between the temporal pattern of the input
signal and that of its internal code. When a neuron receives inputs at its
dendritic spines, it is essentially in a competition with all other neurons
that receive the same inputs. Neurons with internal codes that are
substantially different from the input signals attenuate the signals (by
altering the spine necks of their spines) and delay the propagation of the
signals from the apical tuft to the soma of the neuron (see figure 10). The
neuron with the internal codes most similar to the input signals, will allow
the signals to move freely from the apical dendrite to the soma. The "best
matching neuron", that is, the neuron whose signals past through the apical
dendrite to the soma the fastest, will fire before the other pyramidal
neurons. If the neurons are organized as a competitive circuit (see section
LABEL:ss:lateral_inhibition), the best matching neuron will activate
neighboring inhibitory interneurons, which then inhibit the other pyramidal
neurons in that participated in the competition.

We now know that whenever a neuron fires an action potential, another impulse
from the soma is also generated that propagates backwards through the apical
dendrite of the neuron. This process is also known as Neural backpropagation.
When a backpropagating action potential contacts previously activated
dendritic spines, a superlinear rise in internal calcium levels occurs inside
the dendritic spines. [23, 22]. Neighboring spines that were not activated
prior to the backpropagating signal would be unaffected. I argue that the
backpropagating action potential can be viewed as a mechanism to inform
dendritic spines that their action resulted in a best matching neuron. That
is, it informs the dendritic spines down the apical dendrite that their neuron
won the competition (see figure 11). Every dendritic spine will then modify
their internal code to become slightly more similar to the temporal code that
they received during the competition. In a way the backpropagating signal can
be seen as a message to all the dendritic spines - "We won! Update your codes
slightly so that we can win the next time even faster!" This process can be
viewed as a form of competitive learning [72], which is a variant of Hebbian
learning [73]. The internal codes of each neurons’ dendritic spines change
over time, becoming in the process more specialized towards a specific input.
This in turn makes it more likely for a neuron to win the next competition
again, if the same given input is provided.

It is worth noting that the model so far explains several organizational and
morphological observations: We know that dendritic spines receive most of the
excitatory inputs in neurons, and that practically every dendritic spine has
an excitatory synapse on its head [10, 11, 12]. Excitatory axons prefer to
terminate on spines and almost never connect directly to dendritic shafts (the
body of the dendrite) [12]. This consistent observation makes sense, if we
assume that dendritic spines serve as computational units that try to find
specific temporal patterns within excitatory inputs. Inhibitory axons on the
other hand almost always connect directly to dendritic shafts, rather than on
spines [13]. This also makes sense, if we assume that one of the roles of
inhibitory neurons is to signal the end of a neural competitions. Connecting
directly onto dendritic shafts allows for faster propagation, than propagating
an inhibitory signal through the spine neck. It also explains why most
observed interneurons are essentially spineless [4]. In the Koha model, most
inhibitory neurons serve as signal relayers, not as information processing
units. The competitive circuit also explains why pyramidal neurons posses
about 10 times more local connections with fast-spiking inhibitory neurons,
than with other pyramidal neurons, with most of the connections being
reciprocal (in the layer 2/3 of the rat neocortex at least) [1].

Figure 10: The process of forward propagation within the Koha model. 1 \- The
dendritic spines of the apical tuft receives excitatory signals from a
previous layer. Each spine computes the similarity between the temporal
pattern of the input signal and that of its internal code (see section 2.1). 2
\- The processed signals move from the apical tuft, through the apical
dendrite to the soma. The signal of each neuron will have to cross this
distance. 3 \- The neuron whose propagating signals reach the soma first, will
fire a series of action potentials, which activate surrounding inhibitory
interneurons (shown as circles). The activated interneurons then inhibit all
the other neurons within the competition. Figure 11: The process of backward
propagation within the Koha model. 1 \- The winning neuron fires a series of
action potentials to the next layer, as well as to its neighboring
interneurons (shown as circles), which inhibit the surrounding neurons. In
parallel to that, the winning neuron fires a series of backpropagating action
potentials in the direction of its apical dendrite. 2 \- The propagating
signals move from the winning neuron’s soma, through its apical dendrite, all
the way to the individual dendritic spines at the apical tuft. 3 \- Each
dendritic spine that participated in the competition updates its internal
code, so as to become slightly more similar to the temporal pattern of the
spike train that each dendritic spine received (see section 2.1).

##  3 The Koha model as an associative memory network

The Koha model can be viewed as an associative memory model. The idea of an
associative memory network is to map an input to the most similar internal
pattern. Machine learning models such as the Hopfield Network [45], and the
self organizing map [74] are classical associative memory models. Recently
self-attention-based architectures, such as Transformers [75] have gained wide
popularity in the machine learning space. The newer Modern Hopfield Network
[76], an artificial network that uses the Transofmer’s attention mechanism as
an update rule, is of particular interest. The Modern Hopfield Network shares
many parallels with the Koha hypothesis of biological memory and could be
considered as a continuous implementation of the biological hypothesis. The
"HopfieldLayer" implementation of the Modern Hopfield Network can be defined
as:

$\bm{Z}=softmax\left(\beta\bm{R}\bm{W^{T}_{K}}\right)\bm{W_{V}}$ (1)

The network stores learnable patterns in a weight matrix $\bm{W_{K}}$, also
referred to as the "key weight matrix". The matrix $\bm{R}$ is the input set,
where every row represents a data point. $\beta$ is a parameter for
controlling the "temperature" of the model, whereas $\bm{W_{V}}$ is the "value
weight matrix".

Figure 12: The "HopfieldLayer" configuration of the Modern Hopfield Network.
Reprinted from [76]

From a biological perspective, we can imagine every column of $\bm{W_{K}}$ to
be a neuron with its temporal codes representing the values of the column.
When computing the dot product of a normalized input $\bm{R_{i}}$ with the key
weight matrix $\bm{W_{K}}$, we essentially compute the cosine similarity
between the input and every pattern of the weight matrix. The resulting vector
of cosine similarities represents a score that tells us which neuron is more
similar to the given input. Passing the computed scores to the softmax
function, creates a probability distribution in which patterns more similar to
the input are closer to 1, and dissimilar patterns closer to 0. The described
process can be viewed as a neural competition. $W_{V}$ represents the output
spike train of the neurons, which in themselves would contain precise spike
patterns. If dendritic spines truly posses the ability to scan for precise
spike patterns, than it is natural to assume that every neuron has a specific
temporal "ID", that is, a spike pattern that it transmits. If the activity of
presynaptic neuron is too much out of sync with its original "temporal ID",
the dendritic spines of postsynaptic neurons will dampen the signal, and
therefore weaken the presynaptic neuron’s vote in the dendritic computation of
the postsynaptic neuron’s temporal pattern. In fact, an ingenious experiment
by [77] showed that the temporal order of a spike train emitted by a neuron,
does not get computed in the soma, or even at the axonal initial segment, but
rather in the dendritic tree of the neuron.

##  4 Discussion, Criticism, & Future Work

The current model proposes a mechanism for dendridic spines to scan incoming
signals for temporal patterns, But the exact molecular mechanism remains
unexplained. Synaptopodin’s preferential location at the spine neck, and its
close association with the spine apparatus and the actin cytoskeleton,
suggests that synaptopodin is somehow involved in the rapid plasticity of the
spine’s actin cytoskeleton. One study suggests that synaptopodin could provide
a link between the actin cytoskeleton of the spine neck, and the membrane of
the spine apparatus [38]. Considering that some membrane-bound receptors have
mechanosensitive properties [78], a mechanical interaction between
synaptopodin and the receptor molecules of the spine apparatus might be
possible. If correct, calcium release of the spine apparatus might be
determined by the spine apparatus’ interactions with synaptopodin.

Another detail not explain in depth, is the role of thin spines. Thin spines
in contrast to most mushroom spines do not have a spine apparatus. Because of
this fact, one might argue that the spine apparatus cannot be the only
mechanism for how dendritic spines filter information. Which is correct,
calcium gates within the spine membrane play certainly an important role. It
is however important to note that thin spines have significantly longer necks
than mushroom spines, and are as a result biochemically more isolated. Thin
spines might therefore not play as much of a role in dendritic computations as
mushroom spines do. I argue that in the "neural democracy" of the dendritic
tree, the "vote" of mushroom spines has more value than that of a thin spine.
Considering that mushroom spines can remain stable for a lifetime, while thin
spines can appear and disappear throughout life [8, 9], it would make sense to
weigh the contribution of mushroom spines more.

The Koha model also argues for the existence of a code within dendritic
spines, but it does not describe the physical structure of the hypothetical
code. There are however good candidates. The talin protein, a highly conserved
[79] mechanosensitive cytoskeletal protein in vertebrates, could be encoding
the temporal information that the Koha model suggests exists. In an excellent
paper [80], Goult proposes the MeshCODE hypothesis, in which he argues that
talin has the right properties to serve as a memory molecule. The talin
protein consist of 13 "switches" which can reside in two thermodynamically
stable states. The MeshCODE hypothesis refers to these switch states as "on"
an "off" states and proposes that information can be encoded on the molecule
by either folding, or unfolding the talin switches (see figure 13).
Considering that both talin and synaptopodin are tightly connected to the
actin of the cytoskeleton, a switch controlled mechanical interaction between
talin, actin, and synaptopodin might exist, and could trigger the release of
the spine apparatus’ calcium store down the line. Note however that this is a
hypothesis build on top of another hypothesis and that the correct answer
might be something entirely else.

Figure 13:

Another missing component of the Koha model, is the formation of complex
cells. The described competitive learning process can give rise to simple
cells with simple receptive fields, and even to cells with directional
preferences (by adding skip-connections), but it does not describe, or propose
a biologically plausible way for the formation of "generalized invariances".
That is, it does not describe how neurons become complex cells with invariant
abilities, such as viewpoint invariance. There are several existing hypotheses
for how complex cells form [81], but none seem compelling enough on a
biological level, that is, none show a concrete process for how invariances
are learned naturally. One component that has been purposefully put on the
side in this work is the role of basal dendrites. The whole design of the
proposed competitive learning process depends on the apical dendrite and its
dendritic spines, but ignored the role of basal dendrites, which are in fact
just as important. I argue that by incorporating a few more elements to the
Koha model (such as basal dendrites and the nonliniarities introduced in the
dendritic processing of apical dendrites), one can explain and model the
formation of generalized invariances as well. This is also why the
optimization criteria in the Modern Hopfield Network example was not covered
in detail. The topic of generalized invariances deserves and requires a paper
in itself. How biological neural networks learn invariances, and how
generalized invariances can be used to design better performing artificial
neural networks will be the focus of my subsequent paper.

##  5 Conclusion

This work introduces two novel ideas with widespread implications for the
field of neuroscience:

On a micro level, it argues for the existence of a temporal code within every
dendritic spine. Just as genes are the molecular unit of heredity, the Koha
code is argued to be the unit of memory. It shows how dendritic spines use
these temporal codes to scan for precise spike patterns in their synaptic
inputs. It describes a biologically plausible process for how signal
filtration occurs within spine necks, as well as provides compelling evidence
for the existence of such a mechanism within dendritic spines.

On a macro level, it explains how neurons within competitive circuits can
learn to become pattern detectors, through competitive learning. In this
model, the chance of a neuron to become the "winning neuron" within a
competitive circuit, directly depends on the temporal codes within a neuron’s
dendritic spines.

##  6 Acknowledgements

I would like to thank Taulant Ramabaja and Rita Qarkaxhija for their support,
feedback, and stimulating discussions; without you the ideas in this work
would have never been written. A debt of gratitude is also owed to everyone at
the Institute for Machine Learning at the Johannes Kepler University (JKU) in
Linz. The machine learning theory taught by the professors and researchers at
the JKU served as an invaluable compass in the research and writing of this
paper.

## References

  * [1] Carl Holmgren, Tibor Harkany, Björn Svennenfors, and Yuri Zilberter.  Pyramidal cell communication within local networks in layer 2/3 of rat neocortex.  The Journal of Physiology, 551(Pt 1):139–153, August 2003.
  * [2] D. A. Henze, W. E. Cameron, and G. Barrionuevo.  Dendritic morphology and its effects on the amplitude and rise-time of synaptic signals in hippocampal CA3 pyramidal cells.  The Journal of Comparative Neurology, 369(3):331–344, June 1996\.
  * [3] Nelson Spruston.  Pyramidal neurons: dendritic structure and synaptic integration.  Nature Reviews. Neuroscience, 9(3):206–221, March 2008.
  * [4] Rafael Yuste.  Dendritic Spines.  2010\.  Publisher: The MIT Press.
  * [5] Alan Peters and Ita R. Kaiserman-Abramof.  The small pyramidal neuron of the rat cerebral cortex. The perikaryon, dendrites and spines.  American Journal of Anatomy, 127(4):321–355, 1970.  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/aja.1001270402.
  * [6] K. M. Harris, F. E. Jensen, and B. Tsao.  Three-dimensional structure of dendritic spines and synapses in rat hippocampus (CA1) at postnatal day 15 and adult ages: implications for the maturation of synaptic physiology and long-term potentiation.  The Journal of Neuroscience: The Official Journal of the Society for Neuroscience, 12(7):2685–2705, July 1992.
  * [7] Kalen P. Berry and Elly Nedivi.  Spine Dynamics: Are They All the Same?  Neuron, 96(1):43–55, September 2017.
  * [8] Jennifer Bourne and Kristen M. Harris.  Do thin spines learn to be mushroom spines that remember?  Current Opinion in Neurobiology, 17(3):381–386, June 2007.
  * [9] Anthony J. G. D. Holtmaat, Joshua T. Trachtenberg, Linda Wilbrecht, Gordon M. Shepherd, Xiaoqun Zhang, Graham W. Knott, and Karel Svoboda.  Transient and persistent dendritic spines in the neocortex in vivo.  Neuron, 45(2):279–291, January 2005.
  * [10] E. G. Gray.  Electron Microscopy of Synaptic Contacts on Dendrite Spines of the Cerebral Cortex.  Nature, 183(4675):1592–1593, June 1959.  Number: 4675 Publisher: Nature Publishing Group.
  * [11] J. I. Arellano, A. Espinosa, A. Fairén, R. Yuste, and J. DeFelipe.  Non-synaptic dendritic spines in neocortex.  Neuroscience, 145(2):464–469, March 2007.
  * [12] M. Colonnier.  Synaptic patterns on different cell types in the different laminae of the cat visual cortex. An electron microscope study.  Brain Research, 9(2):268–287, July 1968.
  * [13] Peter Somogyi, Gábor Tamás, Rafael Lujan, and Eberhard H. Buhl.  Salient features of synaptic organisation in the cerebral cortex1Published on the World Wide Web on 3 March 1998.1.  Brain Research Reviews, 26(2):113–135, May 1998.
  * [14] N. Volfovsky, H. Parnas, M. Segal, and E. Korkotian.  Geometry of dendritic spines affects calcium dynamics in hippocampal neurons: theory and experiments.  Journal of Neurophysiology, 82(1):450–462, July 1999.
  * [15] J. Wickens.  Electrically coupled but chemically isolated synapses: dendritic spines and calcium in a rule for synaptic modification.  Progress in Neurobiology, 31(6):507–528, 1988.
  * [16] J. Lisman.  A mechanism for the Hebb and the anti-Hebb processes underlying learning and memory.  Proceedings of the National Academy of Sciences of the United States of America, 86(23):9574–9578, December 1989.
  * [17] C. Koch and A. Zador.  The function of dendritic spines: devices subserving biochemical rather than electrical compartmentalization.  The Journal of Neuroscience: The Official Journal of the Society for Neuroscience, 13(2):413–422, February 1993.
  * [18] William R. Holmes.  Is the function of dendritic spines to concentrate calcium?  Brain Research, 519(1):338–342, June 1990.
  * [19] Sreetama Basu and Raphael Lamprecht.  The Role of Actin Cytoskeleton in Dendritic Spines in the Maintenance of Long-Term Memory.  Frontiers in Molecular Neuroscience, 11, May 2018.
  * [20] Roberto Araya, Jiang Jiang, Kenneth B. Eisenthal, and Rafael Yuste.  The spine neck filters membrane potentials.  Proceedings of the National Academy of Sciences of the United States of America, 103(47):17961–17966, November 2006.
  * [21] Roberto Araya, Volodymyr Nikolenko, Kenneth B. Eisenthal, and Rafael Yuste.  Sodium channels amplify spine potentials.  Proceedings of the National Academy of Sciences of the United States of America, 104(30):12347–12352, July 2007.
  * [22] Rafael Yuste, Ania Majewska, Sydney S. Cash, and Winfried Denk.  Mechanisms of Calcium Influx into Hippocampal Spines: Heterogeneity among Spines, Coincidence Detection by NMDA Receptors, and Optical Quantal Analysis.  Journal of Neuroscience, 19(6):1976–1987, March 1999.  Publisher: Society for Neuroscience Section: ARTICLE.
  * [23] Rafael Yuste and Winfried Denk.  Dendritic spines as basic functional units of neuronal integration.  Nature, 375(6533):682–684, June 1995.  Number: 6533 Publisher: Nature Publishing Group.
  * [24] M. Fischer, S. Kaech, D. Knutti, and A. Matus.  Rapid actin-based plasticity in dendritic spines.  Neuron, 20(5):847–854, May 1998.
  * [25] Anna Dunaevsky, Ayumu Tashiro, Ania Majewska, Carol Mason, and Rafael Yuste.  Developmental regulation of spine motility in the mammalian central nervous system.  Proceedings of the National Academy of Sciences of the United States of America, 96(23):13438–13443, November 1999.
  * [26] E. Korkotian and M. Segal.  Bidirectional regulation of dendritic spine dimensions by glutamate receptors.  Neuroreport, 10(13):2875–2877, September 1999.
  * [27] Jun Noguchi, Masanori Matsuzaki, Graham C.R. Ellis-Davies, and Haruo Kasai.  Spine-Neck Geometry Determines NMDA Receptor-Dependent Ca2+ Signaling in Dendrites.  Neuron, 46(4):609–622, May 2005.
  * [28] Roberto Araya, Tim P. Vogels, and Rafael Yuste.  Activity-dependent dendritic spine neck changes are correlated with synaptic strength.  Proceedings of the National Academy of Sciences of the United States of America, 111(28):E2895–2904, July 2014.
  * [29] Julia Bär, Oliver Kobler, Bas van Bommel, and Marina Mikhaylova.  Periodic F-actin structures shape the neck of dendritic spines.  Scientific Reports, 6(1):37136, November 2016.  Number: 1 Publisher: Nature Publishing Group.
  * [30] Josef Spacek and Kristen M. Harris.  Three-Dimensional Organization of Smooth Endoplasmic Reticulum in Hippocampal CA1 Dendrites and Dendritic Spines of the Immature and Mature Rat.  Journal of Neuroscience, 17(1):190–203, January 1997.  Publisher: Society for Neuroscience Section: Articles.
  * [31] E. Fifková.  A possible mechanism of morphometric changes in dendritic spines induced by stimulation.  Cellular and Molecular Neurobiology, 5(1-2):47–63, June 1985.
  * [32] E. Korkotian and M. Segal.  Fast confocal imaging of calcium released from stores in dendritic spines.  The European Journal of Neuroscience, 10(6):2076–2084, June 1998\.
  * [33] A. H. Sharp, P. S. McPherson, T. M. Dawson, C. Aoki, K. P. Campbell, and S. H. Snyder.  Differential immunohistochemical localization of inositol 1,4,5-trisphosphate- and ryanodine-sensitive Ca2+ release channels in rat brain.  The Journal of Neuroscience: The Official Journal of the Society for Neuroscience, 13(7):3051–3063, July 1993.
  * [34] Alexei Verkhratsky.  Physiology and pathophysiology of the calcium store in the endoplasmic reticulum of neurons.  Physiological Reviews, 85(1):201–279, January 2005.
  * [35] Niklaus Holbro, Åsa Grunditz, and Thomas G. Oertner.  Differential distribution of endoplasmic reticulum controls metabotropic signaling and plasticity at hippocampal synapses.  Proceedings of the National Academy of Sciences of the United States of America, 106(35):15055–15060, September 2009.
  * [36] T. Deller, T. Merten, S. U. Roth, P. Mundel, and M. Frotscher.  Actin-associated protein synaptopodin in the rat hippocampal formation: localization in the spine neck and close association with the spine apparatus of principal neurons.  The Journal of Comparative Neurology, 418(2):164–181, March 2000\.
  * [37] Thomas Deller, Martin Korte, Sophie Chabanis, Alexander Drakew, Herbert Schwegler, Giulia Good Stefani, Aimee Zuniga, Karin Schwarz, Tobias Bonhoeffer, Rolf Zeller, Michael Frotscher, and Peter Mundel.  Synaptopodin-deficient mice lack a spine apparatus and show deficits in synaptic plasticity.  Proceedings of the National Academy of Sciences of the United States of America, 100(18):10494–10499, September 2003.
  * [38] T. Deller, P. Mundel, and M. Frotscher.  Potential role of synaptopodin in spine motility by coupling actin to the spine apparatus.  Hippocampus, 10(5):569–581, 2000.
  * [39] I. Benedeczky, E. Molnár, and P. Somogyi.  The cisternal organelle as a Ca(2+)-storing compartment associated with GABAergic synapses in the axon initial segment of hippocampal pyramidal neurones.  Experimental Brain Research, 101(2):216–230, 1994.
  * [40] Carlos Bas Orth, Christian Schultz, Christian M. Müller, Michael Frotscher, and Thomas Deller.  Loss of the cisternal organelle in the axon initial segment of cortical neurons in synaptopodin-deficient mice.  The Journal of Comparative Neurology, 504(5):441–449, October 2007\.
  * [41] D. H. Hubel and T. N. Wiesel.  Receptive fields of single neurones in the cat’s striate cortex.  The Journal of Physiology, 148(3):574–591, October 1959.
  * [42] D. H. Hubel and T. N. Wiesel.  Receptive fields and functional architecture of monkey striate cortex.  The Journal of Physiology, 195(1):215–243, March 1968.
  * [43] Kunihiko Fukushima.  Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position.  Biological Cybernetics, 36(4):193–202, April 1980.
  * [44] Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel.  Backpropagation Applied to Handwritten Zip Code Recognition.  Neural Computation, 1(4):541–551, December 1989.  Conference Name: Neural Computation.
  * [45] J. J. Hopfield.  Neural networks and physical systems with emergent collective computational abilities.  Proceedings of the National Academy of Sciences, 79(8):2554–2558, April 1982.  Publisher: National Academy of Sciences Section: Research Article.
  * [46] Sepp Hochreiter and Jürgen Schmidhuber.  Long Short-Term Memory.  Neural Computation, 9(8):1735–1780, November 1997.
  * [47] Prof. Gyorgy Buzsaki and Dr. Peter Jonas.  Neural inhibition - Scholarpedia.
  * [48] H. B. Barlow.  Single units and sensation: a neuron doctrine for perceptual psychology?  Perception, 1(4):371–394, 1972.
  * [49] Z. F. Mainen and T. J. Sejnowski.  Reliability of spike timing in neocortical neurons.  Science, 268(5216):1503–1506, June 1995.  Publisher: American Association for the Advancement of Science Section: Reports.
  * [50] Sander M. Bohte.  The evidence for neural information processing with precise spike-times: A survey.  Natural Computing, 3(2):195–206, June 2004.
  * [51] W. Gerstner, A. K. Kreiter, H. Markram, and A. V. Herz.  Neural codes: firing rates and beyond.  Proceedings of the National Academy of Sciences of the United States of America, 94(24):12740–12741, November 1997.
  * [52] S. P. Strong, Roland Koberle, Rob R. de Ruyter van Steveninck, and William Bialek.  Entropy and Information in Neural Spike Trains.  Physical Review Letters, 80(1):197–200, January 1998.  Publisher: American Physical Society.
  * [53] F. Theunissen and J. P. Miller.  Temporal encoding in nervous systems: a rigorous definition.  Journal of Computational Neuroscience, 2(2):149–162, June 1995\.
  * [54] Y. Prut, E. Vaadia, H. Bergman, I. Haalman, H. Slovin, and M. Abeles.  Spatiotemporal structure of cortical activity: properties and behavioral relevance.  Journal of Neurophysiology, 79(6):2857–2874, June 1998.
  * [55] Joy Putney, Rachel Conn, and Simon Sponberg.  Precise timing is ubiquitous, consistent, and coordinated across a comprehensive, spike-resolved flight motor program.  Proceedings of the National Academy of Sciences, 116(52):26951–26960, December 2019.
  * [56] Simon Thorpe.  Spike arrival times: A highly efficient coding scheme for neural networks.  Parallel Processing in Neural Systems and Computers, January 1990\.
  * [57] Daniel A. Butts, Chong Weng, Jianzhong Jin, Chun-I. Yeh, Nicholas A. Lesica, Jose-Manuel Alonso, and Garrett B. Stanley.  Temporal precision in the neural code and the timescales of natural vision.  Nature, 449(7158):92–95, September 2007.  Bandiera_abtest: a Cg_type: Nature Research Journals Number: 7158 Primary_atype: Research Publisher: Nature Publishing Group.
  * [58] Tim Gollisch and Markus Meister.  Rapid Neural Coding in the Retina with Relative Spike Latencies.  Science, 319(5866):1108–1111, February 2008.  Publisher: American Association for the Advancement of Science Section: Report.
  * [59] Alexander Egea-Weiss, Alpha Renner, Christoph J. Kleineidam, and Paul Szyszka.  High Precision of Spike Timing across Olfactory Receptor Neurons Allows Rapid Odor Coding in Drosophila.  iScience, 4:76–83, June 2018.
  * [60] R. Christopher deCharms and Michael M. Merzenich.  Primary cortical representation of sounds by the coordination of action-potential timing.  Nature, 381(6583):610–613, June 1996.  Bandiera_abtest: a Cg_type: Nature Research Journals Number: 6583 Primary_atype: Research Publisher: Nature Publishing Group.
  * [61] W. Bialek, F. Rieke, RR de Ruyter van Steveninck, and D. Warland.  Reading a neural code.  Science, 252(5014):1854–1857, June 1991.  Publisher: American Association for the Advancement of Science Section: Reports.
  * [62] Stefano Panzeri, Rasmus S. Petersen, Simon R. Schultz, Michael Lebedev, and Mathew E. Diamond.  The Role of Spike Timing in the Coding of Stimulus Location in Rat Somatosensory Cortex.  Neuron, 29(3):769–777, March 2001.  Publisher: Elsevier.
  * [63] Emily L. Mackevicius, Matthew D. Best, Hannes P. Saal, and Sliman J. Bensmaia.  Millisecond Precision Spike Timing Shapes Tactile Perception.  Journal of Neuroscience, 32(44):15309–15317, October 2012.
  * [64] Vernon Lawhern, Alexandre A. Nikonov, Wei Wu, and Robert J. Contreras.  Spike Rate and Spike Timing Contributions to Coding Taste Quality Information in Rat Periphery.  Frontiers in Integrative Neuroscience, 5, 2011.  Publisher: Frontiers.
  * [65] Timothée Masquelier and Gustavo Deco.  Learning and Coding in Neural Networks.  pages 513–526. May 2013.
  * [66] Timothée Masquelier, Rudy Guyonneau, and Simon J. Thorpe.  Spike Timing Dependent Plasticity Finds the Start of Repeating Patterns in Continuous Spike Trains.  PLOS ONE, 3(1):e1377, January 2008.  Publisher: Public Library of Science.
  * [67] K. Hayashi, R. Ishikawa, L. H. Ye, X. L. He, K. Takata, K. Kohama, and T. Shirao.  Modulatory role of drebrin on the cytoskeleton within dendritic spines in the rat cerebral cortex.  The Journal of Neuroscience: The Official Journal of the Society for Neuroscience, 16(22):7161–7170, November 1996.
  * [68] Michael Wyszynski, Jerry Lin, Anuradha Rao, Elizabeth Nigh, Alan H. Beggs, Ann Marie Craig, and Morgan Sheng.  Competitive binding of alpha-actinin and calmodulin to the NMDA receptor.  Nature, 385(6615):439–442, January 1997.  Bandiera_abtest: a Cg_type: Nature Research Journals Number: 6615 Primary_atype: Research Publisher: Nature Publishing Group.
  * [69] K. Furukawa, W. Fu, Y. Li, W. Witke, D. J. Kwiatkowski, and M. P. Mattson.  The actin-severing protein gelsolin modulates calcium channel and NMDA receptor activities and vulnerability to excitotoxicity in hippocampal neurons.  The Journal of Neuroscience: The Official Journal of the Society for Neuroscience, 17(21):8178–8186, November 1997.
  * [70] R. K. Carlin, D. C. Bartelt, and P. Siekevitz.  Identification of fodrin as a major calmodulin-binding protein in postsynaptic density preparations.  The Journal of Cell Biology, 96(2):443–448, February 1983.
  * [71] N. Emptage, T. V. Bliss, and A. Fine.  Single synaptic events evoke NMDA receptor-mediated release of calcium from internal stores in hippocampal dendritic spines.  Neuron, 22(1):115–124, January 1999.
  * [72] David E. Rumelhart, James L. McClelland, and San Diego PDP Research Group University of California.  Parallel distributed processing : explorations in the microstructure of cognition.  Cambridge, Mass. : MIT Press, 1986.
  * [73] D. O. Hebb.  The organization of behavior; a neuropsychological theory.  The organization of behavior; a neuropsychological theory. Wiley, Oxford, England, 1949.  Pages: xix, 335.
  * [74] Teuvo Kohonen.  Self-organized formation of topologically correct feature maps.  Biological Cybernetics, 43(1):59–69, January 1982.
  * [75] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin.  Attention Is All You Need.  arXiv:1706.03762 [cs], December 2017.  arXiv: 1706.03762.
  * [76] Hubert Ramsauer, Bernhard Schäfl, Johannes Lehner, Philipp Seidl, Michael Widrich, Thomas Adler, Lukas Gruber, Markus Holzleitner, Milena Pavlović, Geir Kjetil Sandve, Victor Greiff, David Kreil, Michael Kopp, Günter Klambauer, Johannes Brandstetter, and Sepp Hochreiter.  Hopfield Networks is All You Need.  arXiv:2008.02217 [cs, stat], April 2021.  arXiv: 2008.02217.
  * [77] Nicholas Oesch, Thomas Euler, and W. Rowland Taylor.  Direction-selective dendritic action potentials in rabbit retina.  Neuron, 47(5):739–750, September 2005.
  * [78] P. Paoletti and P. Ascher.  Mechanosensitivity of NMDA receptors in cultured mouse central neurons.  Neuron, 13(3):645–655, September 1994.
  * [79] Melissa A. Senetar and Richard O. McCann.  Gene duplication and functional divergence during evolution of the cytoskeletal linker protein talin.  Gene, 362:141–152, December 2005.
  * [80] Benjamin T. Goult.  The Mechanical Basis of Memory – the MeshCODE Theory.  Frontiers in Molecular Neuroscience, 14:21, 2021.
  * [81] Luis M. Martinez and Jose-Manuel Alonso.  COMPLEX RECEPTIVE FIELDS IN PRIMARY VISUAL CORTEX.  The Neuroscientist : a review journal bringing neurobiology, neurology and psychiatry, 9(5):317–331, October 2003.