Statistical Learning in Speech: A Biologically Based Predictive Learning Model
John Rohrlich and Randall C. O’Reilly
Department of Psychology, Computer Science, and Center for Neuroscience
University of California Davis
1544 Newton Ct
Davis, CA 95618
<EMAIL_ADDRESS>August 30, 2024

Thanks to Tom Hazy for comments on a draft version.

R. C. O’Reilly is Chief Scientist at eCortex, Inc., which may derive indirect
benefit from the work presented here.

Supported by: ONR grants N00014-20-1-2578, N00014-19-1-2684/ N00014-18-1-2116,
N00014-18-C-2067, N00014-17-1-2961, N00014-15-1-0033.

All data and materials will be available at
https://github.com/ccnlab/statlearn upon publication.

###### Abstract

Infants, adults, non-human primates and non-primates all learn patterns
implicitly, and they do so across modalities. The biological evidence supports
the hypothesis that the mechanism for this learning is general but
computationally local. We hypothesize that the mechanism itself is predictive
error-driven learning. We build on recent work that proposed a biologically
plausible model of error backpropagation learning which proposes that higher
order thalamic nuclei provide a locale for a temporal difference between top-
down predictions and an actual event outcome. Our neural network based on that
work also models the auditory cortex hierarchy of core, belt and parabelt and
the caudal-rostral axis within regions. We simulated two studies showing
statistical learning in infants, a seminal study using synthesized speech and
a more recent study using human speech. Before simulating these studies the
network was trained on spoken sentences from the TIMIT corpus to emulate
infants experience listening to random speech. The implemented neural network,
learning only by predicting the next brief speech segment, learned in both
simulations to predict in-word syllables better than next-word syllables
showing that prediction could be the basis for word segmentation and thus
statistical learning.

A significant question for cognitive science, especially regarding early human
development, is how we learn from raw sensory experiences. How is it that
infants learn the patterns and regularities in their environment? It is widely
thought that some aspects of this learning are well-captured by the phenomenon
of _statistical learning_ , which has been documented across visual (Fiser
Aslin, 2001, 2002; Kirkham ., 2002) and auditory senses (Saffran ., 1996;
Hauser ., 2001), in non-human primates (Conway Christiansen, 2001; Hauser .,
2001) and across species (Toro Trobalón, 2005), suggesting a fundamental
mechanism. An important feature of statistical learning is that it happens
automatically and without awareness of the product of the learning (Frensch
Rünger, 2003), consistent with a putative role in driving the foundational
learning across sensory domains and a critical role in the development of
language, speech, motor planning, music, and so on. One of the most widely-
studied forms of statistical learning involves infants learning to segment
continuous speech streams based on the co-occurrence statistics of phonemes
(Saffran ., 1996; Graf Estes Lew-Williams, 2015), which is the focus of this
paper.

At a computational level, _predictive learning_ provides an appealing
mechanism driving this phenomenon of statistical learning. Studies going back
to Elman (1990) have shown that a simple recurrent network (SRN) learns
structure by predicting. Studies by Cleeremans McClelland (1991) and Elman
(1991) showed that SRNs, i.e. models that use recent information to predict
what is next, can learn an artificial grammar. Cairns . (1997) found that an
SRN could learn, to an imperfect degree, to segment words from a large corpus
of transcribed English speech. Christiansen . (1998) employing an SRN, and
adding additional cues for better prediction, also modeled speech
segmentation, adding lexical stress and utterance boundaries (two to six
words) to the phoneme information. Two additonal studies modeling segmentation
with SRNs (Giroux Rey, 2009; Plaut Vande Velde, 2017) looked at the role of
chunking and part-words when learning to segment speech. All of these studies
showing that prediction is a good candidate for how infants learn to segment
speech and learn the rules of grammar.

We build on this work to explore a recent proposal for the biological basis of
a closely-related form of predictive learning, based on unique properties of
the corticothalamic circuitry interconnecting the deep layers of the neocortex
and the higher-order thalamic nuclei such as the pulvinar (O’Reilly ., 2021).
In this model, predictive learning starts immediately within the sensory
processing pathways of the brain, e.g., predictive learning of the visual
world takes place within the lower-level visual-specific pathways, as
contrasted with a possible alternative model where this learning is only
operating at a higher, more abstract level. In this paper, we investigate
whether this model can account for relevant empirical data in the domain of
statistical learning in infants in the context of the widely-studied speech
segmentation paradigm (Saffran ., 1996; Graf Estes Lew-Williams, 2015).

In our simulations we found in-word syllable transitions were better predicted
than next-word syllable transitions, with a model that learns solely from
prediction error on the raw auditory speech stream. Given the biological and
modality specificity of our model, it is also important that our model
accurately reflect the known auditory cortical and thalamic pathways and
patterns of connectivity. As such, the neural network we implemented captures
both known auditory neuroscience and corticothalamic computation supporting
predictive learning. Thus, across these different levels of analysis, our
simulations support the proposal that such a sensory-specific, biologically-
based predictive learning mechanism could be the basis for development of word
segmentation in speech, and, by extension, likely other forms of statistical
learning.

###  0.1 Statistical Learning

The term statistical learning was first used by Saffran . (1996) and the first
experiments on infant statistical learning were focused on word segmentation
(Saffran ., 1996; Aslin ., 1998). Those experiments provide evidence that
infants can segment fluent speech on the basis of the co-occurrence
probability between adjacent syllables, a statistical feature called a
transitional probability. There are of course other cues in speech that help
one detect word boundaries but these were absent in these experiments. Since
those first experiments, studies using the same paradigm have demonstrated
statistical learning with tones (Conway Christiansen, 2006), in infants and
adults (Saffran ., 1999; Graf Estes Lew-Williams, 2015), and in newborns (Bulf
., 2011). Statistical learning has also been shown with visual stimuli (Fiser
Aslin, 2002; Bulf ., 2011) and in serial reaction time tasks (Hunt Aslin,
2001). What is the biology supporting this type of learning?

Though there are not many studies addressing the biology underlying
statistical learning they are consistent in showing enhanced activation in
modality specific areas when listening to or viewing repeating, as opposed to
random, streams. Studies of repeating speech streams (Cunillera ., 2009;
Karuza ., 2013; Elmer ., 2021) show this enhanced activation in the superior
temporal gyrus and the superior portion of the ventral premotor cortex when
compared to non-patterned streams. When processing repeating visual sequences
enhanced activation occurs in high-level visual cortex (Turk-Browne ., 2009).
These results suggest that local computations in separate cortical areas are
responsible for statistical learning.

This hypothesis of local computation is further supported by research showing
individual differences in statistical learning across modalities and across
stimuli within modality. If the biology supporting statistical learning was
not in low level modality specific processing we would not expect to see such
differences. Conway Christiansen (2006) found that participants in an
artificial grammar learning paradigm can learn two different sets of
statistical regularities simultaneously when stimuli are from different
modalities, and even within one modality when the two streams differ in a
major perceptual dimension, again supporting the hypothesis that statistical
learning is governed by local computation. Frost . (2015) hold this view and
hypothesize that statistical learning is a set of domain-general computational
principles that work within each modality and within the constraints of that
modality and each brain region. We believe that the most important of these
mechanisms is predictive learning, based on detailed properties of the
neocortex and thalamus, as reviewed next.

###  0.2 Auditory Neuroscience

The biological mechanism proposed by O’Reilly . (2021) is a specific form of
predictive error-driven learning based on distinctive patterns of connectivity
between the neocortex and the higher-order nuclei of the thalamus. For
audition these higher-order nuclei are the anterodorsal and posterodorsal
areas of the medial geniculate body (MGB) (Sherman Guillery, 2006; Usrey
Sherman, 2018). The hypothesis is that learning is driven by the difference
between top-down predictions, generated by numerous weak projections into the
thalamic relay cells (TRCs) in the higher-order nuclei, and the actual
outcomes supplied by sparse, strong driver inputs from lower areas. The driver
inputs originate in cortical layer 5 and are from bursting neurons (5IBs) so
they are brief and periodic, firing roughly every 100 ms, i.e. alpha
frequency. Thus, the prediction error is a temporal difference in activation
states over the higher-order thalamic nuclei, from an earlier prediction to a
subsequent burst of outcome. This temporal difference can drive local synaptic
changes throughout the neocortex, supporting a biologically-plausible form of
error backpropagation that improves the predictions over time (O’Reilly, 1996;
Ackley ., 1985; Hinton McClelland, 1988; Bengio ., 2017; Whittington Bogacz,
2019; Lillicrap ., 2020).

The organization of the human auditory cortex, in part based on what is known
of non-human primate auditory cortex, is generally held to have three
hierarchically organized regions known as the _core, belt_ and _parabelt_
(Kaas Hackett, 2000). Each of these regions is further divided, with the core
regions having at least two areas, the belt region perhaps seven and the
parabelt region two. Connection patterns suggest a forward flow of information
from core to belt to parabelt and also a flow caudal to rostral (Hackett .,
2014; Scott ., 2017). Feedback connections also exist along both axes.

Thalamic input to auditory cortex comes primarily from the MGB. Input to the
core areas, A1 and R, coming from the ventral division of the MGB while belt
and parabelt regions, both caudal and rostral, receive predominantly from the
dorsal divisions of the MGB (Hackett ., 1998; Mothe ., 2006; Lee Sherman,
2010). This difference in thalamic source is central to the predictive
learning mechanism, described in more detail in the next section.

The core areas A1 and R are both tonotopically organized but there are
functional differences. One difference that is well supported is the longer
latency responses in R compared to A1 (Camalier ., 2012; Bendor Wang, 2008).
Another difference in these core areas is in the tracking of amplitude
modulations. Area A1 can track fast acoustic amplitude modulations, on the
order of 20 – 30 ms, whereas R can only track amplitude modulations of 100 ms
or more (Scott ., 2011). Thus, the caudal core auditory cortex responds
quickly to sound onsets and tracks fast amplitude modulations accurately,
whereas the rostral auditory cortex responds more slowly and tracks longer
amplitude modulations (see Jasmin . (2019) for more on rostral-caudal
contributions to auditory processing). These differences are consistent with
the rostral portion being responsible for phoneme analysis (Jasmin ., 2019;
DeWitt Rauschecker, 2012; Camalier ., 2012).

Along the other axis, the hierarchy of core, belt and parabelt, it has been
found that while core fields respond best to tones and noise, belt and
parabelt respond best to more complex stimuli. Recording studies in macaque
have shown that there is a broadening of neural response to wider frequency
ranges moving from core to non-core areas (Rauschecker ., 1995; Recanzone .,
2000). This is consistent with imaging studies in macaques and humans showing
that non-core areas respond best to speech (Binder ., 2000; Liebenthal .,
2005).

##  1 Computational Model

Figure 1 shows the implemented neural network used for the simulations, both
the areas and the most important pathways. The model includes the hierarchy of
core, belt and parabelt regions, both caudal and rostral, and the superior
temporal sulcus (STS) receiving from both the caudal parabelt (CPB) and
rostral parabelt (RPB). Our simplified cortical model has two belt areas
rather than the likely seven (Kaas Hackett, 2000). Moving from core to belt to
parabelt in the model, neuron tuning is broader in both the frequency and the
time dimensions, with projections largely being constrained within the caudal
and rostral areas. However, there are feedforward and feedback projections
between caudal and rostral superficial layers with the feedforward being
stronger than the feedback consistent with the anatomy described in the
previous section. Auditory input is to the core areas A1 and R and is
maintained throughout each 100 ms trial. There is no feedback to the core
areas in this model. All of this is consistent with a traditional model of the
auditory system.

The model goes further by including the higher order thalamic areas (e.g.
RBTh, RPBTh) and an implementation of hypothesized predictive circuity that
drives implicit learning. To make the working of the predictive mechanism more
accessible, the belt, parabelt and superior temporal sulcus cortical areas are
implemented with separate neurons for the superficial layers, (1–4) and the
deeper layers (5–6), which we refer to as corticothalamic (CT suffix). The
thalamic areas are grouped with corresponding cortical layers rather than
shown more anatomically.

The flow of information between superficial layers, deep layers and thalamic
areas is shown in Figure 1. Core areas receive the auditory signal which is
passed via superficial layer pathways to other auditory cortical areas. Each
area’s deep layers, receiving in part from superficial, maintain the context
and via layer 6 project a prediction onto higher order thalamic nuclei. In the
model, prediction takes place during the first 75 ms of an alpha cycle (minus
phase). During the last 25 ms of the alpha cycle layer 5IB cell bursting
occurs (plus phase). This burst from the core areas is the current true input.
The difference in higher order thalamic activation between the prediction and
the actual is the prediction error that is projected back to the cortical
areas which drives learning. Figure 2 shows a snapshot of actual model
activation for a 200 ms period early in training. The “delay” allows time for
the prediction to develop and thus the temporal difference which implicitly
encodes a prediction error. This is a critical point, if the belt and parabelt
were receiving direct thalamic input from the ventral portion of the MGB they
would have the ground truth immediately and there would be no temporal
difference from prediction to outcome.

Speech sequences for pretraining, training and testing were input to the
network 150 ms per trial with a stride of 100 ms. The 100 ms (i.e. alpha
cycle) stride is based on the biology described previously. The 150 ms input
on each trial provides an overlap in processing that prevents artifacts from
abrupt transitions. To encourage invariance, a random duration of silence,
0–25 ms, was added at runtime at the start of each sequence. This was done for
both training and testing (i.e. holdout) sequences.

The 150 ms of sound was processed in time windows of 25 ms, with a 10 ms
overlap, through a Fourier Transform to calculate the power spectrum. This was
processed through a Mel filter bank of 42 filters covering the range of 20 Hz
to 6000 Hz. Finally, the Mel filter-bank output was convolved with 7 gabor
filters which is the input to the A1 and R (i.e. core) layers of the network.

As mentioned in the Auditory Neuroscience section there are processing and
response differences between caudal and rostral core areas and in the many
belt regions as well. We did not attempt to model these differences in detail
as the focus of this work is on the biological mechanism supporting
statistical learning but we recognize that differences in neuron response
timing, amplitude tracking, frequency selectivity and so on will undoubtedly
interact with predictive learning. We did implement one difference, however,
different size gabors for the caudal and rostral input to A1 and R
respectively. The gabor filters used to process the auditory signal to R were
narrower than those for A1 to capture the more fine grained transients
involved in phoneme discrimination consistent with the rostral portion of the
auditory cortex being part of the ventral stream of auditory processing
(DeWitt Rauschecker, 2012; Jasmin ., 2019).

Figure 1: Predictive learning model showing the prediction onto the higher
order thalamic nuclei from cortical layer 6 (e.g. RPBCT $\rightarrow$ RPBTh;
dotted blue arrows), actual auditory signal from cortical layer 5 intrinsic
bursting neurons (e.g. R $\rightarrow$ RPBTh; solid red arrows), and the
prediction error (e.g. RPBTh $\rightarrow$ RPBCT and RPBTh $\rightarrow$ RPB;
dashed green arrows). Caudal predictive pathways (not shown) are the same as
rostral. A few of the typical feedforward and feedback pathways (solid black
arrows) are shown for caudal layers (not shown for rostral layers). Caudal to
rostral feedforward and feedback pathways also not shown. Appendix includes a
full list of projections. Layers: caudal belt (CB), rostral belt (RB), caudal
parabelt (CPB), rostral parabelt (RPB), and superior temporal sulcus (STS)
Figure 2: A 200 ms sample of corticothalamic flow (2 alpha cycles) showing the
prediction activation on the thalamus and the activation from the layer 5IB
cells (showing A1 layers, other areas are similar). This temporal difference
is the error which drives learning. During the first 75 ms of any 100 ms
period (e.g. 1200 ms to 1275 ms) a prediction is projected from deep layers of
cortical areas onto higher order thalamic nuclei. In the later part of the 100
ms period (1275 ms to 1300 ms) cortical layer 5 bursting from the actual
auditory stimulus drives the thalamus. The temporal difference (i.e. the
prediction error) is projected back onto superficial and deep cortical layers,
which can be seen in subtle activation changes when comparing with the
activations at the 75 ms point.

##  2 Simulation 1

Simulation 1 was modeled on the experiments reported in Saffran . (1996) and
Aslin . (1998). In those experiments infants listened to an artificial
language of four nonsense words where each word in the corpus was composed of
three consonant-vowel pairs (CVs). The synthesized CVs were generated at a
constant rate with no acoustic cues to word boundaries. The only clue to word
boundaries was frequency of co-occurrence of CVs. In each experiment a
familiarization phase, a few minutes of listening to the stream of CVs, was
followed by a test phase using isolated words, two from the four word corpus
and two that were composed of the last CV of one word followed by the first
two CVs of another corpus word. Infants responses to the whole-words vs part-
words showed reliable discrimination based on transitional probability. Our
simulation used the same corpus and showed that the model predicts the neural
activations in the next 100 ms time period in line with transitional
probabilities.

##  3 Method

###  3.1 Stimuli

The synthesized sound files from the experiment by Saffran . (1996) were not
available. A small sample of the original speech generation was available
along with comments from one author on how the sound sequence was generated
(J.R. Saffran, personal communication, August 13, 2020). This information
guided the reproduction of the sounds using GnuSpeech (Hill ., 2017). The
corpus was composed of the four trisyllable words, daropi, golatu, pabiku and
tibudo, a total of 12 syllables. To create the speech files randomized lists
of the 12 CVs were passed to GnuSpeech. Each resulting WAV file was edited to
create 12 separate files, one for each CV. The trisyllable words were created
by concatenating from random instances of each CV. Finally, 24 sound files,
all permutations of four words, were generated by concatenating the
trisyllable words. The duration of each sequence was approximately 3730 ms,
with a mean of 3.25 CVs per second. Syllable duration ranged from 167 ms (ti)
to 447 ms (ro). See Appendix for details on the creation of the stimuli.

###  3.2 Train and Test

To simulate infant experience, hearing a wide variety of sentences and voices,
we pretrained on a subset of the TIMIT corpus (Garofolo ., 1993). TIMIT is a
corpus of phonemically and lexically transcribed speech of American English
speakers of different sexes and dialects. We used the 450 “SX” sentences
spoken by the 128 female speakers. Training on the TIMIT corpus was done for
50 epochs with each epoch containing 10 randomly chosen sentences. After every
10 epochs of pretraining we tested the model with the artificial corpus we
would later train on, to have a learning baseline.

Each of the 25 pretraining runs on the TIMIT corpus was followed by training
on the four trisyllable corpus. Training on the trisyllable corpus was for 10
epochs, nine sequences per epoch. The nine sequences (36 trisyllable words)
were chosen randomly each epoch from 20 of the 24 total sequences. Four random
sequences were held back each run for testing. Testing was done after every
epoch for this training. A new random seed was used for each of the 25 runs.

###  3.3 Measuring Performance

If the model is learning to predict it should be able to predict in-word
syllables, which are fully predictable, better than next-word syllables, where
there are three possibilities. We measured prediction during the first 100 ms
of every CV other than the first of each sequence. Specifically, we compared
the thalamic layer activations at the 75 ms point (i.e. the end of the
prediction phase) with the activations of the thalamic layer at the 100 ms
point (i.e. 25 ms after the 5IB neurons begin the periodic burst). The cosine
difference between these values is the measure of prediction.

Though the measurement was done every 100 ms, coinciding with the shift in
input, CVs varied in duration, as they do in all human speech, and
additionally were offset by random silence at the start of each sequence. This
means that the 100 ms period may include the tail of one CV along with the
start of another or that a portion of the start of a CV is missed. This is
true for both in-word and next-word measurements so we looked at the in-word /
next-word comparison rather than any absolute value. Figure 3 is a 1500 ms
sample from a spectrogram showing the 100 ms segments for “golatupabi” with a
0 ms duration of silence at the sequence start. The first 400 ms comprise the
CV “go” though a bit of the CV sound falls beyond 400 ms. The first segment of
“la” includes the tail of “go”. We used a 70% threshold for determining the
CV. Adding silence or sampling from a difference sequence may change the
number of 100 ms segments for a CV. Additionally the same CV will vary in
length for different speakers and also in the synthesized sounds though to a
smaller degree.

Figure 3: A spectrogram showing the first 1500 ms of the 4 word sequence
“golatupabikutibudodaropi”. Prediction was measured on the first 100 ms of
each CV except the first of the sequence, in this case “go”. The second CV,
“la” starts at about 430 ms so we use the 100 ms period from 400–500 ms. If
there had been less than 70 ms of “la” in the 400–500 ms period the next 100
ms would have been considered the first subsegment of “la”. That situation
occurs with the third CV, “tu”, which starts at about 770 ms. We consider the
800–900 ms region the first 100 ms of “tu”.

##  4 Results and Discussion

Data was evaluated using the cosine similarity value for each 100 ms period at
the start of each syllable. A linear mixed model, which handles both fixed and
random effects, was used. The first analysis looked at the test results after
the 5th epoch of training, the point at which the model had “heard” the same
number of words as the infants in the Saffran . (1996) experiment. The model
fixed effects were, the factor of main interest, condition: in-word transition
and next-word transition; layer: the five sets of neurons representing
specific auditory cortical areas. Run, equivalent to one subject, was treated
as a random variable. Comparing the full model to one without the condition
factor showed a significant difference: $\chi^{2}(1)=181.44,p<0.001$,
prediction being better for in-word transitions. Figure 4 shows the result
from Saffran . (1996) experiment and the simulation at a comparable point.
Note that for the Saffran . (1996) experiment the important point is that if
infants listened to either group of words significantly longer it would mean
that they had extracted the crucial sequence information. It was the part-
words they listened to longer. For the simulation the direction of the
difference is important and the difference was in the expected direction, in-
word transitions were predicted better than next-word transitions.

Figure 4: Results from Saffran . (1996) experiment alongside those of
Simulation 1, both having statistically significant differences between
conditions. Note that the measure in the experiment was listening time while
for the simulation it was prediction accuracy. Both charts show the result
after 180 words. Error bars indicate SEs.

Results for the full simulation, rather than epoch 5 in isolation, are shown
in Figure 5. For this analysis epoch was added as a fixed factor, pretraining
and training were analyzed separately. Pretraining on the TIMIT corpus showed
significant learning across epochs, $\chi^{2}(1)=1913.7,p<0.001$, and a
significant effect of condition, $\chi^{2}(1)=219.59,p<0.001$, with next-word
transitions being predicted better than in-word transitions. Analysis of
training on the artificial language showed a significant condition effect
across epochs, $\chi^{2}(1)=1397.8,p<0.001$, as was the case for epoch 5
analyzed by itself. In-word transitions were predicted better than next-word
transitions, as expected. The analysis also showed a significant effect for
layer $\chi^{2}(4)=1547.3,p<0.001$ and the layer by condition interaction
$\chi^{2}(4)=78.63,p<0.001$ (see Figure 6).

For comparison, we also ran the model without the TIMIT pretraining. A
comparable level of prediction performance was achieved after 25 epochs, with
in-word prediction better than next-word prediction, as it was with
pretraining (see Figure 10 in Appendix).

Figure 5: A box plot showing the model predicts in-word transitions better
than next-word transitions. Testing was done every 10 epochs during
pretraining and after each epoch during training. The measure is the
similarity of activations at the end of the “minus” phase of each trial
(prediction) and the actual neuron activations at the end of the “plus” phase.
The infants in the Saffran . (1996) experiment listened to 180 words. The
model was trained on 36 words each epoch. Figure 6: Mean cosine similarity
values by condition and by layer showing higher layers predicting better than
lower and rostral layers predicting better than caudal. There was an
interaction of layer and condition so the planned comparisons were analyzed
separately by condition, see Table 1 for the statistical analysis. The cosine
similarity measures are on the thalamic (Th) area corresponding to each layer.
Layers: caudal belt (CB), rostral belt (RB), caudal parabelt (CPB), rostral
parabelt (RPB), and superior temporal sulcus (STS)

Comparison | df | t-value | p-value
---|---|---|---
STS - RPB | 2485 | 13.84 | p < .001
STS - CPB | 2485 | 7.44 | p < .001
RPB - RB | 2485 | 14.32 | p < .001
CPB - CB | 2485 | 19.99 | p < .001
RPB - CPB | 2485 | 6.40 | p < .001
RB - CB | 2485 | 0.74 | p = 0.46
(a) In-Word Condition

Comparison | df | t-value | p-value
---|---|---|---
STS - RPB | 2485 | 21.37 | p < .001
STS - CPB | 2485 | 9.86 | p < .001
RPB - RB | 2485 | 7.68 | p < .001
CPB - CB | 2485 | 12.72 | p < .001
RPB - CPB | 2485 | 11.50 | p < .001
RB - CB | 2485 | 6.46 | p < .001
(b) Next-Word Condition

Table 1: Table showing the six planned comparisons for Simulation 1 of layers
predicting activations (shown separately for in-word and next-word conditions
because of the interaction). The measure of prediction is the cosine
difference between predicted and actual activations on the thalamus. The
prediction on the thalamus comes from the deep layers of the cortex areas, the
layers with the CT suffix in Figure 1. All layer to layer comparisons were in
the expected direction for the in-word condition, higher layers predicting
better than lower and rostral predicting better than caudal. The results were
the same for the next-word condition with the exception of the RB to CB
comparison which was nonsignificant. Layers: superior temporal sulcus (STS),
rostral parabelt (RPB), caudal parabelt (CPB), rostral belt (RB) and caudal
belt (CB)

By training the model on one corpus, TIMT, and testing on another, our four
nonsense word corpus, we see that the model is learning to predict the
activations from auditory input generally. That next-word transitions were
better predicted than in-word is surprising, we expected no difference, but a
four word corpus is a very small test set. Testing after training on the four
word corpus showed the expected opposite result, better prediction of in-word
(i.e. fully predictable) transitions. All of the six planned comparisons were
in the expected direction for the in-word condition, five of the six for the
next-word condition (see Table 1). The STS layer predicted activations better
than parabelt layers, parabelt layers predicted better than belt layers and
rostral layers predicted better than caudal. This is in line with the two axis
organization of the auditory cortex and will be discussed further in the
General Discussion.

##  5 Simulation 2

Simulation 2 was modeled on experiment 1 reported in (Graf Estes Lew-Williams,
2015). This experiment is different from the Saffran . (1996) experiment,
Simulation 1, primarily in the use of human rather than synthesized speech and
the use of multiple voices, rather than a single voice. Further, it is a more
difficult task because the next-word transition probability was 50% rather
than 33%. In the Graf Estes Lew-Williams (2015) experiment infants, 8- and
10-month-olds, listened to a language of four nonsense words where each word
in the corpus was composed of two syllables. The six minutes of speech heard
by the infants was edited from the recordings of eight female speakers. Each
speaker recorded monotone three-syllable sequences that incorporated all of
the possible coarticulation contexts present in the language and the two
syllable words were created by excising and splicing the middle syllable of
the three-syllable sequences. This technique reduced the chance of speakers
introducing supplemental word boundary cues. To create variation, syllables
from each speaker were spliced together in sequences that ranged from 10 to 20
syllables, a duration of 3 s to 7 s, and the order of speakers was randomized.

##  6 Method

###  6.1 Stimuli

We were able to obtain an original sound file used in the experiments of Graf
Estes Lew-Williams (2015). It was for version “A” of the artificial language
(timay, dobu, gapi, and moku). The counterbalancing language was not
available. The 3 minutes of speech, heard twice in the original experiment,
were subdivided into 18 segments of approximately 10 seconds each.

###  6.2 Train and Test

As in simulation 1, speech sequences were input to the network 150 ms per
trial with a stride of 100 ms. To add variability to the stimuli a short
random duration of silence, 0 - 25 ms, was added at runtime at the start of
each sequence. Sequences were processed in the same manner as for Simulation
1. Each epoch was composed of 6 sequences, about 60 seconds of speech, and the
simulation ran for 10 epochs. The model was run 25 times, holding out three
sound files for testing on each run. Each simulation began with a new random
seed. Performance was measured as in Simulation 1.

##  7 Results and Discussion

Performance of the model was measured in the same manner as for Simulation 1.
The cosine similarity values for each 100 ms period at the start of a syllable
were analyzed with a linear mixed model. We look first at the results of
testing after the 6th epoch of training, the point at which the model had
heard six minutes of speech, the same duration of speech as the infants in the
Graf Estes Lew-Williams (2015) experiment. The model fixed effects were layer:
the five sets of neurons representing specific auditory cortical areas;
condition: in-word transition or next-word transition. Run, equivalent to one
subject, was treated as a random variable. Comparing the full model to one
without the condition factor showed a significant difference:
$\chi^{2}(1)=172.36,p<0.001$. Figure 7 shows the result from Graf Estes Lew-
Williams (2015) experiment and the simulation at a comparable point. In the
Graf Estes Lew-Williams (2015) experiment infants listened significantly
longer to the familiar words (i.e. whole-words). In the simulation in-word
transitions, equivalent to the whole-word condition, were predicted better
than next-word transitions.

Figure 7: Results from Graf Estes Lew-Williams (2015) experiment alongside
those of Simulation 2, both having statistically significant differences
between conditions. Note that the measure in the experiment was listening time
while for the simulation it was prediction accuracy. Both charts show the
result after 6 minutes of “listening” words. Error bars indicate SEs.

Results for the full simulation, rather than epoch 6 in isolation, are shown
in Figure 8. For this analysis epoch was added as a fixed factor, pretraining
and training were analyzed separately. Pretraining on the TIMIT corpus showed
significant learning across epochs, $\chi^{2}(1)=1570.6,p<0.001$, and also a
small effect of condition, $\chi^{2}(1)=10.27,p<.01$, in-word prediction
better than next-word prediction. Analysis of training on the artificial
language showed a significant condition effect across epochs,
$\chi^{2}(1)=1604.9,p<0.001$, as it did looking at epoch 6 separately. In-word
transitions were predicted better than next-word transitions, as expected and
as in Simulation 1. The analysis also showed a significant effect for layer
$\chi^{2}(4)=480.60,p<0.001$ and for the layer by condition interaction
$\chi^{2}(4)=38.28,p<0.001$ (see Figure 9). The planned comparisons, the same
as those for Simulation 1, were in the expected direction (see Table 2) with a
couple of exceptions, STS did not predict better than RPB though it did
predict better than CPB. Also, the CPB / CB comparison showed no difference.
These results are in line with the two axis organization of the auditory
cortex as they were in Simulation 1. For comparison, we also ran the model
without the TIMIT pretraining. A comparable level of prediction performance
was achieved after 20 epochs, with in-word prediction better than next-word
prediction, as it was with pretraining (see Figure 11 in Appendix).

Figure 8: A box plot showing the model predicts in-word transitions better
than next-word transitions. Testing was done every 10 epochs during
pretraining and after each epoch during training. The measure is the
similarity of activations at the end of the “minus” phase of each trial
(prediction) and the actual neuron activations at the end of the “plus” phase.
The infants in the Graf Estes Lew-Williams (2015) experiment listened to 6
minutes of speech. The model was trained with one minute of speech each epoch.
Figure 9: Mean cosine similarity values by condition and by layer showing
parabelt layers predicting better than belt layers and rostral layers
predicting better than caudal. STS prediction was not better than RPB
prediction as expected. There was an interaction of layer and condition so the
planned comparisons were analyzed separately by condition, see Table 2 for the
statistical analysis. The cosine similarity measures are on the thalamic (Th)
area corresponding to each layer. Layers: caudal belt (CB), rostral belt (RB),
caudal parabelt (CPB), rostral parabelt (RPB), and superior temporal sulcus
(STS)

Comparison | df | t-value | p-value
---|---|---|---
STS - RPB | 2485 | 7.28 | p < .001
STS - CPB | 2485 | -2.40 | p = 0.016
RPB - RB | 2485 | 7.84 | p < .001
CPB - CB | 2485 | 5.55 | p < .001
RPB - CPB | 2485 | 9.68 | p < .001
RB - CB | 2485 | 11.96 | p < .001
(a) In-Word Condition

Comparison | df | t-value | p-value
---|---|---|---
STS - RPB | 2485 | 8.19 | p < .001
STS - CPB | 2485 | -1.89 | p = 0.059
RPB - RB | 2485 | 0.43 | p = 0.67
CPB - CB | 2485 | 2.40 | p = 0.016
RPB - CPB | 2485 | 10.07 | p < .001
RB - CB | 2485 | 8.10 | p < .001
(b) Next-Word Condition

Table 2: Table showing the six planned comparisons for Simulation 2 of layers
predicting activations (shown separately for in-word and next-word conditions
because of the interaction). The measure of prediction is the cosine
difference between predicted and actual activations on the thalamus. The
prediction on the thalamus comes from the deep layers of the cortex areas, the
layers with the CT suffix in Figure 1. As expected RPB prediction was better
than RB, CPB prediction better than CB. Both rostral areas predicted better
than caudal. STS did predict better than CPB but not RPB. Layers: superior
temporal sulcus (STS), rostral parabelt (RPB), caudal parabelt (CPB), rostral
belt (RB) and caudal belt (CB)

##  8 General Discussion

The fundamental role of prediction in learning is receiving increasing
attention and support. The simulations reported in this paper show that
prediction is a possible basis for infant word segmentation and statistical
learning more generally. The two simulations replicating experiments in infant
learning both showed that a neural network model based on a biologically
plausible form of error-driven learning and prediction, grounded in the
anatomy of the auditory thalamocortical pathways, learns to predict next
syllable transitions from speech. This learning followed prelearning on a
widely used corpus of varied content and voice that was unrelated to the
artificial language that the model was later trained and tested on. This
prelearning served to simulate infants exposure to a wide variety of voices
and sentences. In both simulations the model predicted in-word syllable
transitions better than less predictable next-word transitions. The infant
experiments reported in Saffran . (1996) and Graf Estes Lew-Williams (2015)
also showed a clear difference between in-word and next-word conditions but
not a reliable direction of effect. In the Saffran . (1996) experiment infants
listened longer to the “unfamiliar” words (i.e. the next-word condition) while
in the Graf Estes Lew-Williams (2015) experiment infants listened longer to
the familiar words in Block 1 and the unfamiliar in Block 2 of testing.
Although our model has no direct equivalent of listening time, we hypothesize
that prediction success and prediction errors contribute to attentional
engagement.

More generally, these results support the hypothesis that predictive learning,
using the hypothesized biologically-based mechanism, underlies statistical
learning in infants. The nearly immediate statistical learning effects after
pretraining suggest that it has established generalizable phoneme-like
representations that provide an efficient basis for subsequent learning about
phoneme-level co-occurrence statistics. These results add to our earlier work
in the visual modality (O’Reilly ., 2021) showing the development of abstract
categorization purely from predictive learning. The hypothesis of predictive
error-driven learning in the thalamocortical loop was first advanced in
O’Reilly . (2014).

A couple of aspects of the specific results reported deserve discussion. In
both simulations in-word transitions were better predicted than next-word
transitions, but the prediction did not reach 100 percent for in-word, though
the transition was fully predictable. We believe the ceiling on prediction is
largely due to measurement on 100 ms boundaries. The varying length of CVs
plus the random silence means that the 100 ms period where the cosine
similarity is measured will rarely coincide with the actual first 100 ms of
the CV auditory signal. We also need to consider the model itself. The
syllables spoken in Simulation 1 had durations in the 167 ms to 447 ms range
while in Simulation 2 the range was 241 ms to 422 ms with the larger portion
of the syllable duration being the vowel sound. This can lead to ambiguity in
predicting. Consider just two words from Simulation 1, “pabiku” and “daropi”.
Using only the last 100 ms or so of the first CV would make “bi” as good a
guess as “ro”. Part of what needs to be learned in learning to predict is what
information is useful and for what period of time. Improving this aspect of
learning is planned for future work.

The interaction of prediction accuracy, layer and condition also deserves some
discussion. In the auditory cortex incoming information generally flows from
core to belt to parabelt and from caudal regions to more rostral, with regions
further along the hierarchy receiving a broader signal in both time and
frequency. Thus individual neurons in the parabelt, for example, have a longer
time horizon to use in predicting than say a belt area neuron. Likewise, the
rostral areas have more information on which to base predictions than caudal
areas. The results from both simulations followed these expectations with
parabelt layers predicting better than belt and rostral layers predicting
better than caudal. STS receives from both caudal and rostral and predicted
better than both in Simulation 1 and better than caudal but not rostral in
Simulation 2. How this aspect of the model matches the biology is something
that might be testable in the future.

Although we attempted to capture major features of auditory cortex, including
both caudal and rostral areas, there are further details that were not fully
captured in our model. There is evidence, for example, that the core rostral
region, R, exhibits longer response latencies than the core caudal region, A1
(Scott ., 2011; Camalier ., 2012; Nourski ., 2014). It has also been reported
that synchronization of spike discharges to modulations of stimulus amplitude
and frequency are integrated over quite different time windows in A1 (20–30
ms) and R (100 ms) (Scott ., 2011). To broadly reflect these differences in
our model, we processed the signal for core areas A1 and R differently, using
narrower filtering in R to capture shorter sound transients that would be
useful in phoneme differentiation.

There are multiple cues involved in learning to segment words, and evidence
suggests that the favored cues change during development, which may provide
further clues into the nature of the learning process. The experiments with
infants that we simulated specifically controlled the speech, both synthesized
and human, to eliminate word boundary cues other than transition probability,
cues such as prosody, relating to linguistic stress, and phonotactic rules,
governing language specific sequences of sounds. Infants use these cues as
well as transition probability. Johnson Jusczyk (2001) pitted prosodic cues
against statistical cues and found that 8-month-olds weighed speech cues more
heavily than statistical cues. Thiessen Saffran (2003) found that 7-month-olds
attended more to statistical cues than to stress cues but 9-month-olds ignored
statistical cues and segmented based on syllable stress. These development
differences could be modeled and further test our predictive learning
hypothesis.

In conclusion, the simulations described in this paper extend the work on
prediction error-driven learning in thalamocortical circuits into the auditory
domain and show that the same mechanism that allows the visual system to learn
to categorize shapes (O’Reilly ., 2021) can do statistical learning in the
auditory circuits. That important learning is done via prediction has been put
forth by Elman (1990); Friston (2005); George Hawkins (2009); Rao Ballard
(1999) to name a few. Our demonstration that predictive learning on raw
auditory signals can result in sensitivity to statistical regularities in the
speech stream that mirrors those of infants provides an initial foundation for
future work on predictive learning of more sophisticated and abstract features
of the speech stream. How far can purely predictive learning on raw sensory
input go toward providing a capable foundation for higher-level speech
comprehension, and motor production? These are questions we are eager to
address in future research.

##  9 Appendix

All of the materials described here, including the speech synthesis, speech
files and the computational model are all available on our github account at:
https://github.com/ccnlab/statlearn. The emergent simulation environment is
at: https://github.com/emer/leabra which contains extensive documentation and
examples that can be run in Python or the Go language. The best place to start
in understanding computationally how the predictive learning model works is
with the FSA model described in O’Reilly, Russin, Zolfaghar Rohrlich (2021),
which is available at:
https://github.com/emer/leabra/tree/master/examples/deep_fsa.

##  10 Stimuli for Simulation 1

The speech sequences for Simulation 1 were synthesized using GnuSpeech using
the following procedure.

We started by generating 12 text strings that were random combinations of the
12 consonant-vowel (CV) pairs . An additional random CV was added to the
beginning and end of each string. Each CV other than the last was followed by
a comma and a space to produce consistent CV sounds. The two extra CVs were
removed after synthesis. Here is an example:

so, pi, tu, la, do, ro, ku, bi, pa, ti, go, bu, da, ru

Each of the strings was passed to GnuSpeech. The settings were standard with
these exceptions:

  * •

voice_name = female

  * •

micro_intonation = 0

  * •

macro_intonation = 0

  * •

intonation_drift = 0

  * •

random_intonation = 0

The MainDictionary file had the following changes or additions:

  * •

bi ’b_ah_i % i

  * •

bu ’b_uu % ca

  * •

da ’d_ar % a

  * •

do ’d_uh_uu % baic

  * •

ku ’k_uu % ca

  * •

la ’l_ar % a

  * •

pi ’p_ah_i % i

  * •

ro ’r_uh_uu % baic

  * •

ti ’t_ah_i % i

  * •

tu ’t_uu % ca

The resulting sound files were processed as follows:

  1. 1.

Each file was imported into Audacity (v2.4.2.0)

  2. 2.

”Sound Finder” (Analysis Menu) was used to locate the sound between silences
(-db 25.0, ”minimum duration” 0.10 s, ”label start” and ”label end” both 0.0 s
and ”add label” at end set to 0 (no). We manually adjusted any ”label”
boundaries that seemed too inclusive or too exclusive.

  3. 3.

Individual sounds were exported (”Export Multiple”)

  4. 4.

The xxx-01.wav and xxx-14.wav files were discarded (i.e. the extra CVs at the
start and end)

  5. 5.

Random instances of each CV were concatenated to form multiple instances of
trisyllable words conforming to the artificial language (i.e. the four words,
daropi, golatu, pabiku and tibudo)

  6. 6.

Randomly chosen instances of the trisyllable words were concatenated to form
the 24 possible combinations of the four words.

  7. 7.

After concatenation each of the 24 wav files was opened in Audacity and the
start and end of each CV was marked. This was done manually by the first
author while looking at the spectrogram and listening to the speech file. The
timing labels for each wav file were exported separately.

##  11 Stimuli for Simulation 2

For the second simulation we began with a wav file of human speech (see Graf
Estes Lew-Williams (2015)). The file was about 3 minutes duration in total.
The file was opened in Audacity and CV starts and ends were labeled as was
done for the synthesized sounds for Simulation 1. The timing information was
exported separately from the wav file.

##  12  Model Layer Sizes and Structure

Figure 1 in the main text shows the general configuration of the model. Table
3 shows the specific sizes of each of the layers, and where they receive
inputs from.

|  | Units | Pools |
---|---|---|---|---
Area | Name | X | Y | X | Y | Receiving Projections
A1 | A1s | 7 | 2 | 6 | 12 |
R | Rs | 7 | 2 | 6 | 12 |
CB | CBs | 5 | 5 | 4 | 5 | A1s CBs RBs CBth CPBs
| CBct | 5 | 5 | 4 | 5 | CBs CBct CBth CPBs CPBct
| CBth | 7 | 2 | 4 | 5 |  A1 CBct CPBct
RB | RBs | 5 | 5 | 4 | 5 | A1s Rs RBs CBs RBth RPBs
| RBct | 5 | 5 | 4 | 5 | RBs RBct RBth RPBs RPBct
| RBth | 7 | 2 | 4 | 5 |  R RBct RPBct
CPB | CPBs | 5 | 5 | 3 | 5 | CBs CPBs RPBs CPBth STSs
| CPBct | 5 | 5 | 3 | 5 | CPBs CPBct CPBth STSct
| CPBth | 7 | 2 | 3 | 5 |  A1 CPBct STSct
RB | RBs | 5 | 5 | 3 | 5 | RBs RPBs CPBs RPBth STSs
| RBct | 5 | 5 | 3 | 5 | RPBs RPBct RPBth STSct
| RBth | 7 | 2 | 3 | 5 |  R RPBct STSct
STS | STSs | 5 | 5 | 3 | 5 | CPBs RPBs STSs STSth
| STSct | 5 | 5 | 3 | 5 | STSs STSct STSth
| STSth | 7 | 4 | 3 | 5 |  A1 R CPBct RPBct
Table 3: Layer sizes, showing numbers of units in one pool and the number of
Pools of such units, along X,Y axes. Each area, other than A1 and R, has three
associated layers: s = superficial layer, ct = corticothalamic layer (context
updated by 51B neurons in same area, shown in bold), th = thalamic layer
(driven by 5IB neurons from associated area, shown in bold).

##  13 Additional Results

Figures showing results for each simulation when the model is _not_ first
trained on the TIMIT sentences.

Figure 10: A box plot of testing results for Simulation 1 on a network that
was not pretrained on the TIMIT sentences. The model learns to predict to a
similar degree as when pretrained but requires more epochs of training.
Testing was done after each epoch during training. The measure is the
similarity of activations at the end of the “minus” phase of each trial
(prediction) and the actual neuron activations at the end of the “plus” phase.
Compare with Figure 5. Figure 11: A box plot of testing results for Simulation
2 on a network that was not pretrained on the TIMIT sentences. The model
learns to predict to a similar degree as when pretrained but requires more
epochs of training. Testing was done after each epoch during training. The
measure is the similarity of activations at the end of the “minus” phase of
each trial (prediction) and the actual neuron activations at the end of the
“plus” phase. Compare with Figure 8.

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${\lambda}^{ij}E_{ij}={\lambda}^{ij}(R_{ij}-\frac{1}{2}{\omega}_{ij}tr(R))=({\lambda}^{ij}-\frac{1}{2}{\lambda}^{rs}{\omega}_{rs}{\omega}^{ij})R_{ij}={\bar{\lambda}}^{ij}R_{ij}$

Integrating by parts while setting as usual $\Box={\omega}^{rs}d_{rs}$, we
obtain:

$(\Box{\bar{\lambda}}^{rs}+{\omega}^{rs}d_{ij}{\bar{\lambda}}^{ij}-{\omega}^{sj}d_{ij}{\bar{\lambda}}^{ri}-{\omega}^{ri}d_{ij}{\bar{\lambda}}^{sj}){\Omega}_{rs}={\sigma}^{rs}{\Omega}_{rs}$

Moreover, suppressing the ”bar ” for simplicity, we have:

$d_{r}{\sigma}^{rs}={\omega}^{ij}d_{rij}{\lambda}^{rs}+{\omega}^{rs}d_{rij}{\lambda}^{ij}-{\omega}^{sj}d_{rij}{\lambda}^{ri}-{\omega}^{ri}d_{rij}{\lambda}^{sj}=0$

As $Einstein$ is a self-adjoint operator (contrary to the Ricci operator), we
have the identities:

$ad(Einstein)=ad(C)\circ ad({\cal{X}})\,\Rightarrow\,Einstein=C\circ
ad({\cal{X}})\,\Rightarrow\,ad({\cal{X}})=Ricci\,\Rightarrow\,{\cal{X}}=ad(Ricci)$

Indeed, $ad(C)=C$ because $C$ is a symmetric matrix and we know that
$ad(Einstein)=Einstein$. Accordingly, the operator $ad(Ricci)$ parametrizes
the $Cauchy$ equations, without any reference to the $Einstein$ operator that
cannot be obtained by any diagram chasing. The three terms after the Dalembert
operator disappear if we add the differential constraints
$d_{i}{\lambda}^{ri}=0$. When $n=4$, we finally obtain the adjoint sequences:

$\begin{array}[]{rccccl}&4&\stackrel{{\scriptstyle
Killing}}{{\longrightarrow}}&10&\stackrel{{\scriptstyle
Ricci}}{{\longrightarrow}}&10\\\ &&&&&\\\
0\leftarrow&4&\stackrel{{\scriptstyle
Cauchy}}{{\longleftarrow}}&10&\stackrel{{\scriptstyle
ad(Ricci)}}{{\longleftarrow}}&10\end{array}$

without any reference to the $Bianchi$ operator and the induced $div$ operator
but the upper sequence is not exact because the CC of the Killing operator are
generated by the Riemann operator, not by the Ricci operator as we saw.
$\Box$

REMARK 8.2: In the opinion of the author of this paper who is not an historian
of sciences but a specialist of mathematical physics interested by the analogy
existing between electromagnetism (EM), elasticity (EL) and gravitation (GR)
by using the conformal group of space-time, it is difficult to imagine that
Einstein could not have been aware of the works of Maxwell and Beltrami on the
foundations of EL and tensor calculus. Indeed, not only they were quite famous
when he started his research work but it must also be noticed that the
phenomenological law of field-matter couplings (piezzoelectricity,
photoelasticity) had been discovered by … Maxwell !.

In order to extend the classical control concepts of poles and zeros to
arbitrary $n$, we need a few more definitions.

DEFINITION 8.3: The symbol $g_{q}$ of a given system $R_{q}\subset J_{q}(E)$
is $g_{q}=R_{q}\cap S_{q}T^{*}\otimes E\subset J_{q}(E)$. With any
differential module $M$ we shall associate the graded module $G=gr(M)$ over
the polynomial ring $gr(D)\simeq K[\chi]$ by setting
$G={\oplus}^{\infty}_{q=0}G_{q}$ with $G_{q}=M_{q}/M_{q+1}$ and we may also
set $g_{q}=G_{q}^{*}=hom_{K}(G_{q},K)$ where the symbol $g_{q+1}$ is defined
by the dual short exact sequences with respect to $K$:

$0\longrightarrow M_{q}\longrightarrow M_{q+1}\longrightarrow
G_{q+1}\longrightarrow 0\hskip 11.38109pt\Longleftrightarrow\hskip
11.38109pt0\longrightarrow g_{q+1}\longrightarrow
R_{q+1}\stackrel{{\scriptstyle{\pi}^{q+1}_{q}}}{{\longrightarrow}}R_{q}\longrightarrow
0$

PROPOSITION 8.4: The Spencer operator $d:R_{q+}\rightarrow T^{*}\otimes R_{q}$
restricts to $\delta:g_{q+1}\rightarrow T^{*}\otimes g_{q}$ up to sign and
more generally to the so-called $\delta$-sequence of Spencer, namely [1, 2]:

$...{\wedge}^{s-1}T^{\otimes}g_{q+r+1}\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}{\wedge}^{s}T^{*}\otimes
g_{q+r}\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}{\wedge}^{s+1}T^{*}\otimes
g_{q+r-1}...$

We denote by $B^{s}_{q+r}(g_{q})=im(\delta)\subseteq
Z^{s}_{q+r}(g_{q})=ker(\delta),H^{s}_{q+r}(g_{q})$ the purely algebraic
coboundary, cocycle and cohomology bundles at ${\wedge}^{s}T^{*}\otimes
g_{q+r}$. The cohomology is vanishing for any $0\leq s\leq n$ and for any
$r\geq 0$ when $R_{q}$ and thus $g_{q}$ is involutive. Also, $g_{q}$ is said
to be $s-acyclic$ if $H^{1}_{q+r}(g_{q})=0,...,H^{s}_{q+r}(g_{q})=0,\forall
r\geq 0$.

Let $R_{q}\subset J_{q}(E)$ be a first order involutive system of order $q$
with no CC of order $1$. The following diagram allows to compute the number
$dim(F_{1})$ of CC of order $2$:

$\begin{array}[]{rcccccccccl}&&0&&0&&0&&0&&\\\
&&\downarrow&&\downarrow&&\downarrow&&\downarrow&&\\\
0&\rightarrow&g_{3}&\rightarrow&S_{3}T^{*}\otimes
T&\rightarrow&S_{2}T^{*}\otimes F_{0}&\rightarrow&F_{1}&\rightarrow&0\\\
&&\downarrow&&\downarrow&&\downarrow&&\parallel&&\\\
0&\rightarrow&R_{3}&\rightarrow&J_{3}(T)&\rightarrow&j_{2}(F_{0})&\rightarrow&F_{1}&\rightarrow&0\\\
&&\downarrow&&\downarrow&&\downarrow&&\downarrow&&\\\
0&\rightarrow&R_{2}&\rightarrow&J_{2}(T)&\rightarrow&J_{1}(F_{0})&\rightarrow&0&&\\\
&&\downarrow&&\downarrow&&\downarrow&&&&\\\ &&0&&0&&0&&&&\end{array}$

We shall apply the previous results to the following inclusion of groups:

$POINCARE\hskip 5.69054ptGROUP\subset CONFORMAL\hskip 5.69054ptGROUP$

that is $10<15$ when $n=4$ and our aim is now to explain why the mathematical
structures of electromagnetism and gravitation only depend on the second order
jets.

With more details, the Killing system $R_{2}\subset J_{2}(T)$ is defined by
the infinitesimal Lie equations in Medolaghi form with the well known Levi-
Civita isomorphism $(\omega,\gamma)\simeq j_{1}(\omega)$ involving the metric
$\omega$ with $det(\omega)\neq 0$ and the corresponding Christoffel symbols
${\gamma}^{k}_{ij}=\frac{1}{2}{\omega}^{kr}({\partial}_{i}{\omega}_{rj}+{\partial}_{j}{\omega}_{ir}-{\partial}_{r}{\omega}_{ij}$.
replacing the partial derivatives of vectors by jet notation, we have the
defining system:

$\left\\{\begin{array}[]{rcl}{\Omega}_{ij}&\equiv&{\omega}_{rj}{\xi}^{r}_{i}+{\omega}_{ir}{\xi}^{r}_{j}+{\xi}^{r}{\partial}_{r}{\omega}_{ij}=0\\\
{\Gamma}^{k}_{ij}&\equiv&{\gamma}^{k}_{rj}{\xi}^{r}_{i}+{\gamma}^{k}_{ir}{\xi}^{r}_{j}-{\gamma}^{r}_{ij}{\xi}^{k}_{r}+{\xi}^{r}{\partial}_{r}{\gamma}^{k}_{ij}=0\end{array}\right.$

We notice that
$R_{2}(\bar{\omega})=R_{2}(\omega)\Leftrightarrow\bar{\omega}=a\,\omega,a=cst,\bar{\gamma}=\gamma$
and refer the reader to [LAP] for more details about the link between this
result and the deformation theory of algebraic structures. We also notice that
$R_{1}$ is formally integrable and thus $R_{2}$ is involutive if and only if
$\omega$ has constant Riemannian curvature along the well known result of L.
P. Eisenhart [ ]. The only structure constant $c$ appearing in the
corresponding Vessiot structure equations is such that $\bar{c}=c/a$ [CST].
The symbol $g_{1}$ with $dim(g_{1})=n(n-1)/2$ is defined by
${\omega}_{rj}{\xi}^{r}_{i}+{\omega}_{ir}{\xi}^{r}_{j}=0$ and we have
$g_{2}=0$. One can find in any GR textbook the fact that the number of
components of the Riemann tensor is
$n^{2}(n+1)^{2}/4-n^{2}(n+1)(n+2)/6=n^{2}(n^{2}-1)/12$, thus $20$ when $n=4$.
As we have the short exact sequence:

$0\longrightarrow{\wedge}^{2}T^{*}\otimes
g_{1}\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}{\wedge}^{3}T^{*}\otimes
T\longrightarrow 0$

the fact that we have also
$dim(H^{2}_{1}(g_{1}))=n^{2}(n-1)^{2}/4-n^{2}(n-1)(n-2)/6=n^{2}(n^{2}-1)/12$
while changing ”$+$” to ”$-$” is definitively proving that the foundations of
Riemannian geometry must be revisited !. The proof can be obtained by applying
the $\delta$-map to the top row of the preceding diagram as it will be done
for the conformal case.

The conformal system ${\hat{R}}_{2}\subset J_{2}(T)$ is defined by the
following infinitesimal Lie equations:

$\left\\{\begin{array}[]{rcl}{\omega}_{rj}{\xi}^{r}_{i}+{\omega}_{ir}{\xi}^{r}_{j}+{\xi}^{r}{\partial}_{r}{\omega}_{ij}&=&2A(x){\omega}_{ij}\\\
{\xi}^{k}_{ij}+{\gamma}^{k}_{rj}{\xi}^{r}_{i}+{\gamma}^{k}_{ri}{\xi}^{r}_{j}-{\gamma}^{r}_{ij}{\xi}^{k}_{r}+{\xi}^{r}{\partial}_{r}{\gamma}^{k}_{ij}&=&{\delta}^{k}_{i}A_{j}(x)+{\delta}^{k}_{j}A_{i}(x)-{\omega}_{ij}{\omega}^{kr}A_{r}(x)\end{array}\right.$

and is involutive if and only if ${\partial}_{i}A-A_{i}=0$ or, equivalently,
if $\omega$ has vanishing Weyl tensor.
Introducing the metric density ${\hat{\omega}}_{ij}={\omega}_{ij}/(\mid
det(\omega\mid)^{\frac{1}{n}}$ and substituting, we obtain the system:

${\hat{\Omega}}_{ij}\equiv{\hat{\omega}}_{rj}{\xi}^{r}_{i}+{\hat{\omega}}_{ir}{\xi}^{r}_{j}-\frac{2}{n}{\hat{\omega}}_{ij}{\xi}^{r}_{r}+{\xi}^{r}{\partial}_{r}{\hat{\omega}}_{ij}=0$

Contracting the first equations by ${\hat{\omega}}^{ij}$ we notice that
${\xi}^{r}_{r}$ is no longer vanishing. It is essential to notice that the
symbols ${\hat{g}}_{1}$ and ${\hat{g}}_{2}$ only depend on $\omega$ and not on
any conformal factor. Hence, we obtain $dim({\hat{g}}_{1})=n(n-1)/2+1$ and
${\hat{g}}_{1}$ is simply defined by
${\omega}_{rj}{\xi}^{r}_{i}+{\omega}_{ir}{\xi}^{r}_{j}-\frac{2}{n}{\omega}_{ij}{\xi}^{r}_{r}=0$
with now $dim({\hat{g}}_{2})=n$. We have proved in [26] that ${\hat{g}}_{3}=0$
when $n\geq 3$, that ${\hat{g}}_{2}$ is $2$-acyclic when $n\geq 4$ and
$3$-acyclic when $n\geq 5$ (These results have been checked by computer
algebra up to $n=5$ [39]).

When $n=4$ and
${\hat{g}}_{3}=0\Rightarrow{\hat{g}}_{4}=0\Rightarrow{\hat{g}}_{5}=0$ in the
conformal case, we have the commutative diagram with exact vertical long
$\delta$-sequences but the left one:

$\begin{array}[]{rcccccccccl}&&0&&0&&0&&&&\\\
&&\downarrow&&\downarrow&&\downarrow&&&&\\\
0&\rightarrow&{\hat{g}}_{3}&\rightarrow&S_{3}T^{*}\otimes
T&\rightarrow&S_{2}T^{*}\otimes{\hat{F}}_{0}&\rightarrow&{\hat{F}}_{1}&\rightarrow&0\\\
&&\hskip 5.69054pt\downarrow\delta&&\hskip 5.69054pt\downarrow\delta&&\hskip
5.69054pt\downarrow\delta&&&&\\\
0&\rightarrow&T^{*}\otimes{\hat{g}}_{2}&\rightarrow&T^{*}\otimes
S_{2}T^{*}\otimes T&\rightarrow&T^{*}\otimes
T^{*}\otimes{\hat{F}}_{0}&\rightarrow&0&&\\\ &&\hskip
5.69054pt\downarrow\delta&&\hskip 5.69054pt\downarrow\delta&&\hskip
5.69054pt\downarrow\delta&&&&\\\
0&\rightarrow&{\wedge}^{2}T^{*}\otimes{\hat{g}}_{1}&\rightarrow&{\wedge}^{2}T^{*}\otimes
T^{*}\otimes
T&\rightarrow&{\wedge}^{2}T^{*}\otimes{\hat{F}}_{0}&\rightarrow&0&&\\\
&&\hskip 5.69054pt\downarrow\delta&&\hskip
5.69054pt\downarrow\delta&&\downarrow&&&&\\\
0&\rightarrow&{\wedge}^{3}T^{*}\otimes T&=&{\wedge}^{3}T^{*}\otimes
T&\rightarrow&0&&&&\\\ &&\downarrow&&\downarrow&&&&&&\\\
&&0&&0&&&&&&\end{array}$

$\begin{array}[]{rcccccccccl}&&&&0&&0&&&&\\\ &&&&\downarrow&&\downarrow&&&&\\\
&&0&\rightarrow&80&\rightarrow&90&\rightarrow&10&\rightarrow&0\\\
&&\downarrow&&\hskip 5.69054pt\downarrow\delta&&\hskip
5.69054pt\downarrow\delta&&&&\\\
0&\rightarrow&16&\rightarrow&160&\rightarrow&144&\rightarrow&0&\\\ &&\hskip
5.69054pt\downarrow\delta&&\hskip 5.69054pt\downarrow\delta&&\hskip
5.69054pt\downarrow\delta&&&&\\\
0&\rightarrow&42&\rightarrow&96&\rightarrow&54&\rightarrow&0&&\\\ &&\hskip
5.69054pt\downarrow\delta&&\hskip 5.69054pt\downarrow\delta&&\downarrow&&&&\\\
0&\rightarrow&16&=&16&\rightarrow&0&&&&\\\ &&\downarrow&&\downarrow&&&&&&\\\
&&0&&0&&&&&&\end{array}$

A diagonal snake chase proves that ${\hat{F}}_{1}\simeq H^{2}({\hat{g}}_{1})$.
However, we obtain at once $dim(B^{2}({\hat{g}}_{1}))=16$ but, in order to
prove that the number of components of the Weyl tensor is $42-32=10$ or,
equivalently, to prove that $dim(Z^{2}({\hat{g}}_{1}))=42-16=26$, we have to
prove that the last map $\delta$ in the left Weyl $\delta$-sequence is
surjective, a result that it is almost impossible to prove in local
coordinates. Let us prove it by means of circular diagram chasing in the
preceding commutative diagram as follows. Lift any
$a\in{\wedge}^{3}T^{*}\otimes T$ to $b\in{\wedge}^{2}T^{*}\otimes T^{*}\otimes
T$ because the vertical $\delta$-sequence for $S_{3}T^{*}$ is exact. Project
it by the symbol map ${\sigma}_{1}({\hat{\Phi}})$ to
$c\in{\wedge}^{2}T^{*}\otimes{\hat{F}}_{0}$. Then, lift $c$ to $d\in
T^{*}\otimes T\otimes{\hat{F}}_{0}$ that we may lift backwards horizontally to
$e\in T^{*}\otimes S_{2}T^{*}\otimes T$ to which we may apply $\delta$ to
obtain $f\in{\wedge}^{2}T^{*}\otimes T^{*}\otimes T$. By commutativity, both
$f$ and $b$ map to $c$ and the difference $f-b$ maps thus to zero. Finally, we
may find $g\in{\wedge}^{2}T^{*}\otimes{\hat{g}}_{1}$ such that $b=g+\delta(e)$
and we obtain thus $a=\delta(g)+{\delta}^{2}(e)=\delta(g)$, proving therefore
the desired surjectivity. Going one step further, we let the reader discover
the following result, found in 2016 but still not acknowledged today !:
WHEN $n=4$, THE WEYL TENSOR WITH $10$ COMPONENTS HAS ONLY $9$ GENERATING
BIANCHI-LIKE CC OF ORDER $2$;
In order to prove that both classical and conformal differential geometry must
be entirely revisited, let us prove that the analogue of the Weyl tensor is
made by a third order operator when $n=3$ which is also neither known nor
nacknowledged today. As before, we shall proceed by diagram chasing as the
local computation can only be done by using computer algebra and does not
provide any geometric insight (See [39] for the details).

$\begin{array}[]{rcccccccccl}&&0&&0&&0&\\\
&&\downarrow&&\downarrow&&\downarrow&\\\
0&\rightarrow&{\hat{g}}_{4}&\rightarrow&S_{4}T^{*}\otimes
T&\rightarrow&S_{3}T^{*}\otimes{\hat{F}}_{0}&\rightarrow&{\hat{F}}_{1}&\rightarrow&0\\\
&&\downarrow&&\downarrow&&\downarrow&&\\\
0&\rightarrow&T^{*}\otimes{\hat{g}}_{3}&\rightarrow&T^{*}\otimes
S_{3}T^{*}\otimes T&\rightarrow&T^{*}\otimes
S_{2}T^{*}\otimes{\hat{F}}_{0}&\rightarrow&0&&\\\
&&\downarrow&&\downarrow&&\downarrow&&&\\\
0&\rightarrow&{\wedge}^{2}T^{*}\otimes{\hat{g}}_{2}&\rightarrow&{\wedge}^{2}T^{*}\otimes
S_{2}T^{*}\otimes T&\rightarrow&{\wedge}^{2}T^{*}\otimes
T^{*}\otimes{\hat{F}}_{0}&\rightarrow&0&&\\\
&&\downarrow&&\downarrow&&\downarrow&&&\\\
0&\rightarrow&{\wedge}^{3}T^{*}\otimes{\hat{g}}_{1}&\rightarrow&{\wedge}^{3}T^{*}\otimes
T^{*}\otimes
T&\rightarrow&{\wedge}^{3}T^{*}\otimes{\hat{F}}_{0}&\rightarrow&0&\\\
&&\downarrow&&\downarrow&&\downarrow&&&\\\
0&\rightarrow&{\wedge}^{4}T^{*}\otimes T&=&{\wedge}^{4}T^{*}\otimes
T&\rightarrow&0&&&\\\ &&\downarrow&&\downarrow&&&&&\\\ &&0&&0&&&&&\end{array}$

$\begin{array}[]{rcccccccccl}&&&&0&&0&\\\ &&&&\downarrow&&\downarrow&\\\
&&0&\rightarrow&45&\rightarrow&50&\rightarrow&5&\rightarrow&0\\\
&&&&\downarrow&&\downarrow&&\\\
&&0&\rightarrow&90&\rightarrow&90&\rightarrow&0&&\\\
&&\downarrow&&\downarrow&&\downarrow&&&\\\
0&\rightarrow&9&\rightarrow&54&\rightarrow&45&\rightarrow&0&&\\\
&&\downarrow&&\downarrow&&\downarrow&&\\\
0&\rightarrow&4&\rightarrow&9&\rightarrow&5&\rightarrow&0&\\\
&&\downarrow&&\downarrow&&\downarrow&&&\\\ &&0&&0&&0&&&\end{array}$

We have indeed: $3\,translations\,+\,3\,rotations\,+\,1\,dilatation\
+3\,elations\,=10$ parameters.
We obtain from a chase ${\hat{F}}_{1}=H^{2}({\hat{g}}_{1})$ and the totally
unexpected formally exact sequences on the jet level are thus, showing in
particular that second order CC do not exist:

$0\rightarrow{\hat{R}}_{3}\rightarrow J_{3}(T)\rightarrow
J_{2}({\hat{F}}_{0})\rightarrow 0\,\,\Rightarrow\,\,0\rightarrow 10\rightarrow
60\rightarrow 50\rightarrow 0$ $0\rightarrow{\hat{R}}_{4}\rightarrow
J_{4}(T)\rightarrow J_{3}({\hat{F}}_{0})\rightarrow{\hat{F}}_{1}\rightarrow
0\,\,\,\Rightarrow\,\,0\rightarrow 10\rightarrow 105\rightarrow 100\rightarrow
5\rightarrow 0$ $0\rightarrow{\hat{R}}_{5}\rightarrow J_{5}(T)\rightarrow
J_{4}({\hat{F}}_{0})\rightarrow
J_{1}({\hat{F}}_{1})\rightarrow{\hat{F}}_{2}\rightarrow
0\,\,\,\Rightarrow\,\,0\rightarrow 10\rightarrow 168\rightarrow 175\rightarrow
20\rightarrow 3\rightarrow 0$

We obtain the minimum differential sequence, which is nervertheless not a
Janet sequence:

$0\longrightarrow{\hat{\Theta}}\longrightarrow
T\underset{1}{\stackrel{{\scriptstyle{\hat{\cal{D}}}}}{{\longrightarrow}}}{\hat{F}}_{0}\underset{3}{\stackrel{{\scriptstyle{\hat{\cal{D}}}_{1}}}{{\longrightarrow}}}{\hat{F}}_{1}\underset{1}{\stackrel{{\scriptstyle{\cal{D}}_{2}}}{{\longrightarrow}}}{\hat{F}}_{2}\longrightarrow
0\,\,\,\Rightarrow\,\,\,0\longrightarrow{\hat{\Theta}}\longrightarrow
3\underset{1}{\stackrel{{\scriptstyle{\hat{\cal{D}}}}}{{\longrightarrow}}}5\underset{3}{\longrightarrow}5\underset{1}{\longrightarrow}3\rightarrow
0$

with $\hat{\cal{D}}$ the conformal Killing operator and vanishing Euler-
Poincaré characteristic $3-5+5-3=0$. We have proved very recently (to appear
soon) that the $5\times 5$ operator ${\hat{\cal{D}}}_{1}$ is self-adjoint.

As a byproduct, we end this paper with the following fundamental diagram $II$
first presented in $1983$ [18] but still not yet acknowledged as it only
depends on the Spencer $\delta$-map, explaining both the splitting vertical
sequence on the right existing in Riemannian geometry and the vector bundle
isomorphism ${\it Ricci}\simeq S_{2}T^{*}$. Indeed, the diagonal chase
providing it could not be even imagined by using classical methods because its
involves Spencer $\delta$-cohomology with the standard notations
$B=im(\delta),Z=ker(\delta),H=Z/B$ for coboundary, cocycle, cohomology at
${\wedge}^{s}T^{*}\otimes g_{q+r}$ when $g_{q+r}$ is the $r$-prolongation of a
symbol $g_{q}$. It is important to notice that all the bundles appearing in
this diagram only depend on the metric $\omega$ but not on any conformal
factor.

$\begin{array}[]{rcccccccccl}&&&&&&&&&&\\\ &&&&&&&&0&&\\\
&&&&&&&&\downarrow&&\\\ &&&&&&0&&Ricci&&\\\ &&&&&&\downarrow&&\downarrow&&\\\
&&&&0&\longrightarrow&Z^{2}_{1}(g_{1})&\longrightarrow&Riemann&\longrightarrow&0\\\
&&&&\downarrow&&\downarrow&&\downarrow&&\\\
&&0&\longrightarrow&T^{*}\otimes{\hat{g}}_{2}&\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}&Z^{2}_{1}({\hat{g}}_{1})&\longrightarrow&Weyl&\longrightarrow&0\\\
&&&&\downarrow&&\downarrow&&\downarrow&&\\\
0&\longrightarrow&S_{2}T^{*}&\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}&T^{*}\otimes
T^{*}&\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}&{\wedge}^{2}T^{*}&\longrightarrow&0&&\\\
&&&&\downarrow&&\downarrow&&&&\\\ &&&&0&&0&&&&\\\ &&&&&&&&&&\end{array}$

When $n=4$, we have explained in recent books [8, 39] and papers [28, 32],
that the horizontal lower sequence is splitting because
$a_{i,j}=\frac{1}{2}(a_{i,j}+a_{j,i})+\frac{1}{2}(a_{i,j}-a_{j,i})$ and
provides an isomorphism $T^{*}\otimes{\hat{g}}_{2}\simeq T^{*}\otimes
T^{*}\simeq S_{2}T^{*}\oplus{\wedge}^{2}T^{*}$ which can be locally described
by $(R_{ij},F_{ij})$ in which $(R_{ij})$ is the GR part and $(F_{ij})$ the EM
part as a unification of gravitation and electromagnetism. Accordingly, the
bundle splitting $Riemann\simeq Ricci\oplus Weyl$ is thus only depending on
the second order jets of conformal transformations, contrary to the philosophy
of GR today. We finally notice that such a result is contradicting the
mathematical foundations of classical gauge theory relating EM to $U(1)$ while
allowing to understand the confusion done by E. Cartan and followers between
”curvature alone” ($F_{1}$) and ”curvature + torsion”
($C_{2}={\wedge}^{2}T^{*}\otimes R_{2}$).

9) CONCLUSION:

We have already proved in our book ”Lie Pseudogroups and Mechanics” (1988)
that the group foundation of elasticity pioneered by E. and F. Cosserat (1909)
was just described by adding to the non-linear Spencer sequence (1972) a
convenient variational calculus. As a crucial conclusion and similarly to
classical elasticity, even if the initial background is non-linear, the
resulting stress/couple-stress equations are linear. Accordingly, in order to
parametrize these equations, one just needs to refer to infinitesimal Lie
equations and the three corresponding canonical linear differential sequences
that can be constructed, namely the gauge sequence, the Janet sequence and the
Spencer sequence, though only the last one is useful.

The main result of this paper, coming from unavoidable arguments of
homological algebra, is that the Cosserat couple-stress equations are just
described by the formal adjoint of the first Spencer operator while the formal
adjoint of the second Spencer operator just describes a possible
parametrization. Accordingly, the parametrization of the Cosserat couple-
stress equations is first order while the parametrization of the classical
Cauchy stress equations is second order as it comes from dualizing second
order CC described by the Riemann operator. When $n=4$, we have also proved
that the use of differential double duality applied to the Spencer sequence
for the conformal Lie pseudogroup allows to unify the Cosserat, Maxwell and
Einstein equations, hence elasticity, electromagnetism and gravitation, along
the dream of H. Weyl and that such a result only depends on the second ordrer
jets called ”elations” by E. Cartan in $1920$. However, we have proved that
the mathematical theory of gravitational waves is not coherent with double
differential duality, a fact explaining why Einstein himself has not been very
confident in their existence all along his life.

Nevertheless, the most surprising result is that the same homological
arguments, namely the systematic use of the adjoint operators, is also
providing the need to revisit the mathematical foundations of control theory.
Indeed, when a differential field $K$ having $n$ commuting derivations is
given together with two finitely generated differential extensions $L$ and $M$
of $K$, an important problem in differential algebra is to exhibit a common
differential extension $N$ in order to define the new differential extensions
$L\cap M$ and the smallest differential field $(L,M)\subset N$ containing both
$L$ and $M$. Such a result allows to generalize the use of complex numbers in
classical algebra. Similarly, having now two finitely generated differential
modules $L$ and $M$ over the non-commutative ring ring
$D=K[d_{1},...,d_{n}]=K[d]$ of differential operators with coefficients in
$K$, we may look for a differential module $N$ containing both $L$ and $M$ in
order to define $L\cap M$ and $L+M$. This is exactly the situation met in
linear or non-linear OD or PD control theory by selecting the inputs and the
outputs among the system variables. However, in many recent books and papers,
we have shown that controllability was a built-in property of a control
system, not depending on the choice of inputs and outputs among the system
variables. Another purpose of this paper is thus to revisit the mathematical
foundations of control theory by showing the specific importance of the two
previous problems and the part plaid by $N$ in both cases for the
parametrization of the control system. The essential tool will be the study of
differential correspondences, a modern name for what was called Bäcklund
problem during the last century, namely the study of elimination theory for
groups of variables among systems of linear or nonlinear OD or PD equations.
The difficulty is to revisit differential homological algebra by using non-
commutative localization. Finally, when $M$ is a $D$-module, this paper is
using for the first time the fact that the system $R=hom_{K}(M,K)$ is a
$D$-module for the Spencer operator acting on sections, avoiding thus
behaviors, trajectories and signal spaces. We finally hope that the many
explicit applications presented will be used in the future as test examples
for computer algebra.

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