# LEPTONIC $\bm{g-2}$ IN 2HDM

FRANCISCO J. BOTELLA  FERNANDO CORNET-GOMEZ 111Speaker and corresponding
author<EMAIL_ADDRESS>CARLOS MIRÓ  MIGUEL NEBOT  Departament de
Física Teòrica and IFIC, Universitat de València-CSIC
Instituto de Física Corpuscular, C/Catedrático José Beltrán, 2. E-46980
Paterna, Spain.

###### Abstract

The experimental observations of the electron and muon anomalous magnetic
moment present discrepancies with respect to the Standard Model predictions. A
class of flavor conserving Two Higgs Doublet model, stable under
renormalization, that is capable of explaining both anomalies simultaneously
is presented. This model can also explain an excess observed by ATLAS in
$\sigma(pp\to\mathrm{S})_{[\text{ggF}]}\times\text{Br}(\mathrm{S}\to\tau^{+}\tau^{-})$.

##  1 Introduction

In this proceeding, we review and extend the works presented in Refs.[1, 2, 3]
regarding the simultaneous explanation of the electron and muon $g-2$
anomalies. Furthermore, we present a preliminary result concerning an excess
in the di-tau channel of a heavy Higgs with a mass of $m_{S}\sim 400$ GeV
observed by ATLAS [4]. The most recent determination of the anomalous magnetic
moment of the muon [5, 6] enforces the long-standing discrepancy (4.2$\sigma$)
between the Standard Model (SM) prediction [7] and the experimental
observation

$\delta a_{\mu}^{\rm Exp}=a_{\mu}^{\rm Exp}-a_{\mu}^{\rm SM}=(2.5\pm
0.6)\times 10^{-9}\,.$ (1)

There is a recent lattice calculation [8] of the Hadronic Vacuum Polarization
(HVP) contribution that shifts the SM prediction to the experimental value,
solving the existing discrepancy. However, it was shown in Ref. [9] that
solving the discrepancy with the HVP contribution would create a tension of
the same size in the electroweak precision data fits. Until this lattice
calculation is crosschecked by other collaborations, we assume the SM
prediction of the $(g-2)_{\mu}$ as the one provided by the White Paper 2020
[7], which is in tension with the experimental observation.

Regarding the electron, the situation is not so clear. There are two different
observations that are incompatible between them and that also deviate slightly
from the SM prediction. The observation from atomic recoil using Cesium
provides [10]

$\delta a_{e}^{\rm Exp,Cs}=-(8.7\pm 3.6)\times 10^{-13}\,,$ (2)

that imply a 2.4 $\sigma$ deviation, while the one using Rubidium [11]

$\delta a_{e}^{\rm Exp,Rb}=(4.8\pm 3.0)\times 10^{-13}\,,$ (3)

only deviates 1.6$\sigma$ but in the opposite direction.

We will focus our attention on discussing $\delta a_{e}^{\rm Exp,Cs}$, since
it is the hardest to explain with scalar mediators due to the different sign
with respect to $\delta a_{\mu}^{\rm Exp}$, but we will also show some results
of the $\delta a_{e}^{\rm Exp,Rb}$ analysis.

a) Current situation with the experimental measurements and the different
determinations of the muon $(g-2)$. Plot from L. Lellouch talk.

b) Allowed $\delta a_{\mu}^{\rm Exp}$ vs. $\delta a_{e}^{\rm Exp}$ regions in
the different analyses. The same color coding will be used in the following
plots.

Figure 1: Electron and muon anomalies studied in this proceedings.

##  2 The model

The 2HDM is a minimal extension of the SM where just a second scalar doublet
of $\text{SU}(2)_{L}$ is added. In this class of models, a new flavor
structure, ${\rm N}_{f}$, appears in the Yukawa sector:

$\begin{array}[]{rc@{\,}c@{\,}l}\mathscr{L}_{\rm
Y}&=&-&\frac{\sqrt{2}}{v}\bar{Q}_{L}\left(H_{1}{\rm M}_{d}+H_{2}{\rm
N}_{d}\right){d}_{R}-\frac{\sqrt{2}}{v}\bar{Q}_{L}\left(\tilde{H}_{1}{\rm
M}_{u}+\tilde{H}_{2}{\rm N}_{u}\right){u}_{R}\\\
&&-&\frac{\sqrt{2}}{v}\bar{L}_{L}\left(H_{1}{\rm M}_{\ell}+H_{2}{\rm
N}_{\ell}\right){\ell}_{R}+\text{h.c.}\,,\\\ \end{array}$ (4)

where ${\rm M}_{f}$ are the diagonal fermion mass matrices. It is well-known
that in the 2HDM, the matrices ${\rm N}_{f}$ are a source of Flavor Changing
Neutral Couplings (FCNC), since in general, they are not simultaneously
diagonalizable with the mass matrices. Thus, the inclusion of a second Higgs
doublet constitutes a challenge since FCNC are very constrained experimentally
but it also opens the possibility of explaining observations that the SM
cannot accommodate. To avoid FCNC we will apply general Flavor Conservation
(gFC), that is, we impose that both Yukawa matrices are diagonalizable
simultaneously. On the one hand, the quark sector of our model corresponds to
a Type I 2HDM where the flavor matrices are proportional by the ratio of the
two doublets vacuum expectation values, ${\rm N}_{q}=t_{\beta}^{-1}{\rm
M}_{q}$. On the other hand, the lepton matrix ${\rm N}_{\ell}$ is chosen to be
arbitrary and diagonal (gFC)

${\rm N}_{\ell}=\begin{pmatrix}n_{e}&0&0\\\ 0&n_{\mu}&0\\\
0&0&n_{\tau}\end{pmatrix}\,,$ (5)

which is one-loop stable under Renormalization Group Evolution as was proved
in Ref. [12]. As the $g-2$ is a CP-conserving observable, we are not
interested in including new sources of CP-violation thus the $n_{\ell}$ are
defined as reals, and the scalar potential reads

$\begin{array}[]{rc@{\,}c@{\,}l}V&=&&m_{11}^{2}H_{1}^{\dagger}H_{1}+m_{22}^{2}H_{2}^{\dagger}H_{2}-\left(m_{12}^{2}H_{1}^{\dagger}H_{2}+\text{h.c.}\right)\\\
&&+&\frac{1}{2}\bar{\lambda}_{1}\left(H_{1}^{\dagger}H_{1}\right)^{2}+\frac{1}{2}\bar{\lambda}_{2}\left(H_{2}^{\dagger}H_{2}\right)^{2}+\bar{\lambda}_{3}\left(H_{1}^{\dagger}H_{1}\right)\left(H_{2}^{\dagger}H_{2}\right)+\bar{\lambda}_{4}\left(H_{1}^{\dagger}H_{2}\right)\left(H_{2}^{\dagger}H_{1}\right)\,,\end{array}$

where the term $\left(m_{12}^{2}H_{1}^{\dagger}H_{2}+\text{h.c.}\right)$
softly breaks the $\mathbb{Z}_{2}$ symmetry that generates the Type I in the
quark sector. This soft-breaking allows us to reach scalar masses above 1 TeV
and values of $t_{\beta}$ greater than 8 that are otherwise forbidden by
perturbativity constraints. The fact that the scalar potential in eq. (2)
conserves CP leads to a scalar sector formed by two scalar fields,
$\mathrm{h}$, and $\mathrm{H}$, one pseudoscalar $\mathrm{A}$ and two charged
particles $\mathrm{H}^{\pm}$. In the so-called scalar alignment limit, the
couplings of $\mathrm{h}$ coincide with those of the SM Higgs
($\mathrm{h}_{125}$).

##  3 Analysis

The main goal of the work presented here is to explain both the electron and
muon anomalies simultaneously. We can split the prediction for the anomalous
magnetic moment, $a_{\ell}^{\rm Th}$, in the SM prediction, $a_{\ell}^{\rm
SM}$, plus the new contribution coming from the 2HDM, $\delta a_{\ell}$, as

$a_{\ell}^{\rm Th}=a_{\ell}^{\rm SM}+\delta a_{\ell},$ (6)

and we can parametrize the New Physics (NP) contribution as

$\delta a_{\ell}=K_{\ell}\,\Delta_{\ell},\qquad
K_{\ell}=\frac{1}{8\pi^{2}}\left(\frac{m_{\ell}}{v}\right)^{2}.$ (7)

To reproduce the experimental observations we need to impose $\delta
a_{\ell}=\delta a_{\ell}^{\text{Exp}}$ what fixes the values of
$\Delta_{\ell}$ in the different scenarios to

$\Delta_{\mu}\simeq 1,\qquad\Delta_{e}^{\rm Cs}\simeq-16,\qquad\Delta_{e}^{\rm
Rb}\simeq 9\,.$ (8)

It is well-known that both the one and two loops contributions can be
relevant. The one-loop contribution contains diagrams with neutral and charged
mediators ($\mathrm{H}$, $\mathrm{A}$ and $\mathrm{H}^{\pm}$)

$\Delta_{\ell}^{(1)}=|n_{\ell}|^{2}\left(\frac{I_{\ell\mathrm{H}}}{m_{\mathrm{H}}^{2}}-\frac{I_{\ell\mathrm{A}}-2/3}{m_{\mathrm{A}}^{2}}-\frac{1}{6{m_{\mathrm{H}^{\pm}}}^{2}}\right)$
(9)

where the loop function reads

$I_{\ell S}\simeq-\frac{7}{6}-2\ln\left(\frac{m_{\ell}}{m_{S}}\right)\,.$ (10)

Among all the possible two-loop diagrams, the dominating ones are the Barr-Zee
which contribute with

$\Delta_{\ell}^{(2)}=-\frac{2\alpha}{\pi}\,\frac{n_{\ell}}{m_{\ell}}\,F\,,$
(11)

where $F$, a quantity that is independent of $\ell$, is defined as

$F=\frac{t_{\beta}^{-1}}{3}\left[4(f_{t\mathrm{H}}+g_{t\mathrm{A}})+(f_{b\mathrm{H}}-g_{b\mathrm{A}})\right]+\frac{\text{Re}\left(n_{\tau}\right)}{m_{\tau}}(f_{\tau\mathrm{H}}-g_{\tau\mathrm{A}})+\frac{\text{Re}\left(n_{\mu}\right)}{m_{\mu}}(f_{\mu\mathrm{H}}-g_{\mu\mathrm{A}}).$
(12)

The two-loop functions $f(z)$ and $g(z)$ are defined as

$\begin{array}[]{rc@{\,}c@{\,}l}f(z)&=\frac{z}{2}\int_{0}^{1}dx\,\frac{1-2x(1-x)}{x(1-x)-z}\,\ln\left(\frac{x(1-x)}{z}\right)\,,\quad
g(z)&=\frac{z}{2}\int_{0}^{1}dx\,\frac{1}{x(1-x)-z}\,\ln\left(\frac{x(1-x)}{z}\right)\,.\end{array}$
(13)

In addition to the $g-2$ anomalies, a big amount of available observables are
included in the fit in order to constrain the parameter space:

  * •

Scalar sector: the scalar potential is imposed to be bounded from below.
Perturbative unitarity of $2\to 2$ scattering amplitudes is also required. As
it was mentioned before, the $\mathbb{Z}_{2}$ symmetry is broken by having
$\mu_{12}^{2}\neq 0$ to allow heavy scalars and $t_{\beta}>8$. Electroweak
precision data is included through the oblique parameters.

  * •

The Yukawa couplings must remain perturbative:

$\frac{|n_{f}|}{v}\leq\mathcal{O}(1)\quad\Rightarrow\quad|n_{f}|\lesssim
250\,\text{GeV}$ (14)

  * •

Signal strengths data constrain the prediction of cross section times
branching ratio of the scalar $\mathrm{h}$. The current data favors the
alignment limit.

  * •

Lepton flavor universality including leptonic and semileptonic decays.

  * •

Flavor constraints: neutral meson-mixing and $b\to s\gamma$.

  * •

LEP data, in particular $e^{+}e^{-}\to\ell^{+}\ell^{-}$.

  * •

LHC direct searches of extra scalar and pseudoscalar particles.

For further details on the predictions in this model of the different
observables and the experimental values used in the fits see Refs. [2, 3].

##  4 Results

A global fit to all the observables mentioned above is performed. To do so, a
$\chi^{2}$-like function is sampled using Markov chain MonteCarlo techniques.
The allowed regions of the parameter space are presented as 2D region plots
where from darker to lighter the 1, 2 and 3$\sigma$ regions of
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{Min}$ are shown. In Fig. 2 the allowed
regions for $\text{Re}\left(n_{\mu}\right)$ vs $\text{Re}\left(n_{e}\right)$
are presented for the three different scenarios studied. In blue, we use
$\delta a_{e}^{\rm Exp,Cs}$ in eq. (2). In this case, the electron anomaly has
the opposite sign to that of the muon anomaly. This usually entails a
challenge for scalar mediators but in our case, we have a decoupling between
electron and muon Yukawa couplings allowing us to explain both simultaneously.
In red, we show the case with $\delta a_{e}^{\rm Exp,Rb}$ in eq. (3). Finally,
in green, we present the situation where the electron anomaly is an average
between the Cesium and Rubidium determinations. As we expected, the most
constrained scenario is the analysis with $\delta a_{e}^{\rm Exp,Cs}$ and for
that reason, we also find it the most attractive to study in-depth.

Figure 2: Allowed regions of $\text{Re}\left(n_{\mu}\right)$ vs
$\text{Re}\left(n_{e}\right)$ for the tree studied scenarios concerning
$\delta a_{e}^{\rm{Exp}}$. We follow the color scheme presented in Fig. 1.

Figure 3 contains some relevant correlations between the couplings,
$t_{\beta}$ and the scalar masses for the $\delta a_{e}^{\rm Exp,Cs}$
scenario. On the left, $\text{Re}\left(n_{e}\right)$ vs $t_{\beta}$ is shown.
We see that smaller values of $t_{\beta}$ require smaller couplings. It is
also worth mentioning that for large $t_{\beta}$ the quark contribution to the
$F$ function in eq. (12) is very suppressed and then, the tau one becomes
dominant. This allows us to have a negative sign for Re($n_{e}$) (if we have
negative Re($n_{\tau}$)). On the center plot $m_{\mathrm{H}}$ is presented as
a function of $t_{\beta}$. We can see that the larger $t_{\beta}$ gets, the
lighter the scalars are. On the right hand side, $m_{\mathrm{H}}$ is plotted
as a function of $m_{\mathrm{H}^{\pm}}$. We know that in this class of models,
the oblique parameters constraint imposes that at low masses, there must be a
degeneracy between $m_{\mathrm{H}^{\pm}}$ and $m_{\mathrm{H}}$ or
$m_{\mathrm{A}}$. We can see in this figure that we get this below 1 TeV. We
can see as well that in the case where $\mathrm{H}^{\pm}$ must be degenerate
with $\mathrm{A}$ there is a lower bound of around 500 GeV. This prevents the
electron anomaly to be explained at one-loop since eq. (9) forces to have
$\text{Re}\left(n_{e}\right)=m_{\mathrm{A}}$ and this would violate
perturbativity constraints ($\text{Re}\left(n_{e}\right)\lesssim 250$ GeV). On
the other hand, the muon one-loop solution would require
$\text{Re}\left(n_{\mu}\right)=m_{\mathrm{A}}/4$, and given the fact that
there are available regions with $m_{\mathrm{H}}<1$ TeV, perturbativity
constraints do not suppose a problem in this case. We can identify then two
scenarios, one where both anomalies are two-loop explained, in the heavy
region, and a lighter region in mass where the electron remains two-loop but
the muon is one-loop instead.

Figure 3: Correlation among some of the parameters in the analysis with the
the Cesium determination for the electron anomaly.

Figure 4 contains some relevant information regarding the tau lepton. In the
left plot, $\text{Re}\left(n_{\tau}\right)$ is presented versus $t_{\beta}$.
As we can see, the tau coupling is almost unconstrained, even more in the low
$t_{\beta}$ region. For that reason, we asked ourselves if there was another
anomaly regarding taus that this model could give an explanation to. In
purple, we show the region of the parameter space that is in agreement with
the observed excess by ATLAS in the direct search of a heavy Higgs
($m_{\mathrm{S}}\sim 400$ GeV) in the di-tau channel [4]. In the center and
right figures we can see the gluon-gluon fusion (ggF) production cross section
of a scalar $\mathrm{H}$ (center) and a pseudoscalar $\mathrm{A}$ (right),
times the branching ratio to $\tau^{+}\tau^{-}$ with respect to the mediator
mass. The black dotted line corresponds to the expected limit while the solid
one is the observed limit by ATLAS. The purple regions of Fig. 4 correspond to
scenarios where this excess is explained with $\mathrm{H}$ or $\mathrm{A}$ in
the model presented here.

Figure 4: Some correlations involving tau leptons. The regions where the
excess in
$\sigma(pp\to\mathrm{S})_{[\text{ggF}]}\times\text{Br}(\mathrm{S}\to\tau^{+}\tau^{-})$
is explained are plot in purple.

##  5 Summary and conclusions

We have presented a flavor conserving two Higgs doublet model that shares the
quark sector with a type I or X but that is general flavor conserving in the
lepton sector. This model has been proved to be one-loop stable under
renormalization. This framework is capable of accommodating both the electron
and muon ($g-2$) anomalies, including the different determinations of the
electron one. In the region where the scalars are heavier than 1 TeV, the two-
loop contributions for both observables are dominating. On the other hand, for
lighter scalar and $t_{\beta}\gg 1$, the muon anomaly is one-loop dominated
while the electron remains two-loop due to perturbativity and universality
constraints. This model is also capable of explaining the small excess
observed by ATLAS in
$\sigma(pp\to\mathrm{S})_{[\text{ggF}]}\times\text{Br}(\mathrm{S}\to\tau^{+}\tau^{-})$
at around 400 GeV.

## Acknowledgments

We want to thank the organizers of the 56th Rencontres the Moriond for setting
up the conference and for giving FCG the opportunity of sharing our work. The
authors acknowledge support from Spanish Agencia Estatal de
Investigación–Ministerio de Ciencia e Innovación (AEI-MICINN) under grants
PID2019-106448GBC33 and PID2020-113334GB-I00/AEI/10.13039/501100011033
(AEI/FEDER, UE) and from Generalitat Valenciana under grant PROMETEO 2019-113.
FCG acknowledges the MICINN, Spain (Grant BES-2017-080070) for supporting his
work. CM is supported by Conselleria de Innovación, Universidades, Ciencia y
Sociedad Digital from Generalitat Valenciana (grant ACIF/2021/284). MN is
funded by the GenT Plan from Generalitat Valenciana under project
CIDEGENT/2019/024.

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(neuron 45) learned silence class, we can find that less type 101 $Y$ neurons
are grown to learn the silence feature since the type 100 $Y$ neurons learned
volume information help to distinguish silence from other phoneme frames.

Figure 17:

The $Y$ neurons’ locations during the procedure of growth are visualized in
Fig. 17A to Fig. 17D. Fig. 17A: early stage; Fig. 17B and Fig. 17C:
intermediate stage’s situations with Fig. 17C later than Fig. 17B; Fig. 17D:
final stage. The little blue points, evenly distributed in the skull space,
are the somas of glial cells, and the red points are the somas of type 101 $Y$
neurons. The green points are somas of type 100 $Y$ neurons. We can see
clearly that the $Y$ neurons explore from the center to the whole space. The
$Y$ neurons are distributed unevenly because each $Y$ neuron births at a
different time and the number of neurons with a similar learning experience is
different.

##  IX Conclusions

As far as we know, presented here is the first developmental method that is
backed by the theory of emergent Universal Turing Machines. Theoretically,
Strong AI is based on a solid mathematical foundation of logic — Universal
Turing Machines — and a solid mathematical foundation of optimality — maximum
likelihood.

Strong AI seems to be practical too: Because of the high dimensionality of
emergent internal features and the long lifetime, the seemingly
computationally intractable problem of batch optimization has been efficiently
addressed through maximum likelihood estimation of the new DN-2. The
optimality is under the conditions of incremental learning, limited learning
experiences, and limited resources. There are no iterations needed. DN-2
enables a fluid hierarchy of international representation. This fluid
hierarchy hopefully is useful for us to reconsider the human handcrafted
hierarchies of brain regions (e.g., those of Van Essen [7] and others but do
not explain the cross-modal plasticity of Sur and coworkers [32]) which are
still popular in neuroscience. The time complexity of a network like DN-2 is
linear in time if the number of neurons is a large number but constant, as
proved in Weng 2012 [38]. Namely, only a constant amount of computation is
needed at each time instance — suitable for an imbedded computer hardware
running in real time and we do not need to worry that it will slow down its
per-frame response regardless of tasks. The slowdown will take place when the
task is complex and the skills of the learner are still weak. This is caused
by the lack of confidence on the part of the learner.

The experimental result seems to support that in practice this mechanism
improves the generalization powers of alternative methods that use input and
action patterns only. The visualization of the internal representations
supports the theory of fluid hierarchy.

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## Appendix

###  IX-A Proof of Lemma 1

Proof. We break down the proof into two steps. Step 1 shows that the firing
pattern in each zone is actually a binary ML estimator of the current input
conditioned on $N(t)$. Step 2 shows that after learning the weights are then
incrementally updated according to ML estimation.

Step 1, case 1 (new neuron enters learning stage at time $t+1$): When a new
neuron $g$ enters learning stage for the first time, it memorizes the input
vector $\mathbf{p}(t+1)$ perfectly. This neuron also suppresses other neurons
inside its $\bar{g}$ from firing (eliminating noises), thus becoming the ML
estimator of the current input $\mathbf{p}(t+1)$.

Step 1, case 2 (no new neurons are added at time $t+1$): For each neuron $g\in
Y,Z$, consider its inhibition zone at time $t$: $\bar{g}(t)$. $c(g,t)$ is the
number of neurons in its inhibition zone $\bar{g}(t)$. The normalized weights
of these $c$ neurons can be denoted as
$(\mathbf{w}_{1},\mathbf{w}_{2},...,\mathbf{w}_{c(g,t)})$.

Then we can define $c(g,t)$ Voronoi regions $R_{j},j=1,2,...,c(g,t)$ in
$L(g,t)$ (receptive field of $g$ at time $t$, which is a linear subspace of
$X\times Y\times Z$), where each $R_{j}$ contains all $\mathbf{p}\in L(g,t)$
that are closer to $\mathbf{w}_{j}$ than to other $\mathbf{w}_{i}$:

$R_{j}=\\{\mathbf{p}|j=\arg\max_{1\leq i\leq
c}\mathbf{w}_{i}\cdot\mathbf{p}\\},\quad j=1,2,...,c$

Given observation $\mathbf{p}(t+1)$, the conditional probability density
$h(\mathbf{p}(t+1)|L(g,t),W(g,t))$ is zero if $\mathbf{p}(t+1)$ falls out of
the Voronoi region of neuron $g$:

$f\\{\mathbf{p}(t+1)|L(g,t),W(g,t)\\}=\begin{cases}f_{i}\\{\mathbf{p}(t+1)|L(g,t),W(g,t)\\},\\\
\quad\quad\quad\textnormal{if}\quad\mathbf{p}(t+1)\in R_{i};\\\
0,\quad\textnormal{otherwise}\end{cases}$ (41)

where $f_{i}\\{\mathbf{p}(t+1)|L(g),W(g)\\}$ is the probability density within
$R_{i}$. Note that the distribution of $f_{i}\\{\mathbf{p}(t+1)|L(g),W(g)\\}$
within $R_{i}$ is irrelevant as long as it integrates to 1.

Given $\mathbf{p}(t+1)$, the ML estimator for the binary vector
$\mathbf{y}(t)$ (also $\mathbf{z}(t)$) needs to maximize
$f\\{\mathbf{p}(t+1)|N(t)\\}$, which is equivalent to finding the set of
firing neuron $n(t+1)$:

$\displaystyle n(t+1)$ $\displaystyle=$ $\displaystyle\\{\arg\max_{g\in
Y,Z}f\\{\mathbf{p}(t+1)|N(t)\\}\\}$ (42) $\displaystyle=$
$\displaystyle\\{g|g=\arg\max_{g\in Y,Z}f\\{\mathbf{p}(t+1)|L(g,t),W(g,t)\\}$
$\displaystyle=$
$\displaystyle\\{g|g=\arg\max_{j\in\bar{g}(t)}\mathbf{w}(j)\cdot\mathbf{p}(t+1)\\}$

since finding the ML estimator in Eq. (41) is equivalent to finding the
Voronoi region to which $\mathbf{p}(t)$ belongs to. This is exactly what the
$Y$ zone does, supposing $k=1$ for top-$k$ competition for each neuron’s
competition zone.

Step 2: Here we are going to show the statistical efficiency of the LCA
learning rule of $W(g)$, where $g\in n(t)$ (meaning that $g$ is among the
firing neurons at time $t$).

Eq. (11) in [41] shows that the candid version of LCA is actually an
incremental estimation of the average of the input that triggers firing in
that specific neuron:

$\displaystyle\mathbf{w}(g,t+1)$ $\displaystyle=$
$\displaystyle\frac{a(g,t)-1}{a(g,t)}\mathbf{w}(g,t)+\frac{1}{a(g,t)}\mathbf{p}(t+1)$
(43) $\displaystyle=$
$\displaystyle\frac{1}{a(g,t)}\Sigma_{i=1}^{a(g,t)}\mathbf{p}(t_{i})$

where $\mathbf{w}(g,t)$ is the weight vector $\mathbf{w}(g)$ at time $t$.
$a(g,t)$ is the age of neuron $g$ at time $t$. $t_{i},i=1,2,...,a(g,t)$ are
the times where neuron $g$ wins dynamic top-$k$ competition and fires.

Same as the proof of optimality in LCA [41], statistical estimation theory
reveals that for many distributions (e.g., Gaussian and exponential
distributions), the sample mean is the most efficient estimator of the
population mean. This follows directly from Th. 4.1, p.429-430 in [15], which
states that under some regularity conditions satisfied by many distributions
(such as Gaussian and exponential distributions), the maximum likelihood
estimator (MLE) $\hat{\boldsymbol{\theta}}$ of the parameter vector
$\boldsymbol{\theta}$ is asymptotically efficient, in the sense that its
asymptotic covariance matrix is the Cramer-Rao information bound (the lower
bound) for all unbiased estimators via convergence in probability to a normal
distribution:

$\sqrt{n}(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta})\xrightarrow{p}N\\{0,I(\boldsymbol{\theta})^{-1}\\}$
(44)

in which the Fisher information matrix $I(\boldsymbol{\theta})$ is the
covariance matrix of the score vector

$\\{(\partial
f(\mathbf{x},\boldsymbol{\theta}))/(\partial\theta_{1}),...,(\partial
f(\mathbf{x},\boldsymbol{\theta}))/(\partial\theta_{k})\\}.$

And $f(\mathbf{x},\boldsymbol{\theta})$ is the probability density of random
vector $\mathbf{x}$ if the true parameter value is $\boldsymbol{\theta}$. The
matrix $I(\boldsymbol{\theta})^{-1}$ is called information bound since under
some regularity constraints, any unbiased estimator
$\tilde{\boldsymbol{\theta}}$ of the parameter vector $\boldsymbol{\theta}$
satisfies $\text{cov}(\tilde{\boldsymbol{\theta}}-\boldsymbol{\theta})\geq
I(\boldsymbol{\theta})^{-1}/n$ (see, e.g., [15], p. 428 or [40], p. 203-204]).

Thus, as Weng et al. showed in [41], the LCA learning algorithms learns the
optimal weights in the sense of maximum likelihood estimation.

Combining step 1 and step 2 we would have:

$\theta(t+1)=\max_{\theta}(t)f\\{\mathbf{p}(t)|N(t-1)\\}$

###  IX-B Proof of Theorem 1

Proof. Intuitively, although $f^{\prime}\\{\mathbf{p}(t+1)|{\cal
X}_{0}^{t},{\cal Z}_{0}^{t},\Gamma\\}$ is in a different format compared to
the $f\\{\mathbf{p}(t+1)|N(t)\\}$ in Lemma 1, the two probability functions
are acting the same as $N(t)$ is determined by ${\cal X}_{t},{\cal Z}_{t}$ and
$\Gamma$. As there is no random weight initialization in DN-2, two DN-2
equipped learning agents would be exactly identical given the same hyper-
parameter $\Gamma$ and learning experience ${\cal X}_{t},{\cal Z}_{t}$. To
formally prove this we are going to recursively use the conclusion of Lemma 1.

At $t+1$, we can reuse Eq. (42) from Lemma 1 together with the condition of
incremental learning:

$\displaystyle\theta(t+1)$ $\displaystyle=$
$\displaystyle\max_{\theta(t)}f\\{\mathbf{p}(t+1)|N(t)\\}$ (45)
$\displaystyle=$ $\displaystyle\max_{\theta(t)}f\\{\mathbf{p}(t+1)|{\cal
L}(N(t-1),\mathbf{x}(t),\mathbf{z}(t),\Gamma)\\}$ $\displaystyle=$
$\displaystyle\max_{\theta(t)}f^{1}\\{\mathbf{p}(t+1)|N(t-1),{\cal
X}_{t}^{t},{\cal Z}_{t}^{t},\Gamma\\}$ $\displaystyle=$
$\displaystyle\max_{\theta(t)}f^{2}\\{\mathbf{p}(t+1)|N(t-2),{\cal
X}_{t-1}^{t},{\cal Z}_{t-1}^{t},\Gamma\\}$ $\displaystyle=$ $\displaystyle...$
$\displaystyle=$
$\displaystyle\max_{\theta(t)}f^{t}\\{\mathbf{p}(t+1)|N(0),{\cal
X}_{0}^{t},{\cal Z}_{0}^{t},\Gamma\\}$ $\displaystyle=$
$\displaystyle\max_{\theta(t)}f^{\prime}\\{\mathbf{p}(t+1)|{\cal
X}_{0}^{t},{\cal Z}_{0}^{t},\Gamma\\}$

where ${\cal L}$ is the learning function of the network, and $f^{i}$ is the
probability density function of the current input $\mathbf{p}$, conditioned on
the network $i$ time steps ago and the learning experience from $t-i$ to $t$.

## CLAIMS

What is claimed is

  1. 1.

Claims 21: (currently amended) An improvement to a developmental network
implemented in computer hardware, the developmental network comprising at
least one neural network having a plurality of neurons organized into a
hierarchy of levels comprising an $X$ area associated with sensory
information, a $Z$ area associated with motor information, and a hidden $Y$
area between the $X$ area and the $Z$ area, the improvement comprising:
each neuron has an associated location in a computer simulated physical space;
and
the internal representation of the hierarchy of the $X$ area, $Y$ area, and
$Z$ area is a fluid hierarchy.

  2. 2.

Claim 22 (currently amended): The improvement of Claim 21, wherein each neuron
has an inhibition zone that is a zone for neuronal competition that determines
whether the neuron fires.

  3. 3.

Claim 23 (currently amended): The improvement of Claim 22, wherein each neuron
associated with at least one negative neuron.

  4. 4.

Claim 24 (currently amended): The improvement of Claim 23, wherein each neuron
includes one or more of excitatory and inhibitory connections, the excitatory
connections correspond to a receptive field of the neuron and the inhibitory
connections correspond to a receptive field of one or more negative neurons
that inhibit the neuron, such that the inhibition zone for any particular
positive neuron changes according to a training or exploration experience of
the particular neuron.

  5. 5.

Claim 25 (currently amended): The improvement of claim 24, wherein each neuron
is classified as presynaptic, post-synaptic, or both; and wherein each post-
synaptic neuron uses the strengths of connections from all the input neurons
to its negative neuron to identify the boundary of its competition zone.

  6. 6.

Claim 26 (currently amended): The improvement of claim 25, wherein each neuron
has a pre-response and each post-synaptic neuron determines whether to fire by
comparing its pre-response with all the pre-responses of all neurons in its
competition zone.

  7. 7.

Claim 27 (currently amended): The improvement of claim 26, wherein each post-
synaptic neuron has a pre-response and wherein each post-synaptic neuron makes
the determination of whether to fire by determining whether the pre-response
of the neuron is valued among the top-$k$ values in the competition zone,
where $k$ is a percentage number.

  8. 8.

Claim 28 (currently amended): The improvement of claim 22, wherein each neuron
in the Y area has a neuronal weight associated with a pre-synaptic neuron and
a post-synaptic neuron and wherein the post-synaptic neuron is updated by a
Hebbian-like mechanism.

  9. 9.

Claim 29 (previously amended): The improvement of claim 21, wherein each
associated location is a 3D location and the computer-simulated physical space
is a 3D space.

  10. 10.

Claim 30 (previously amended): The improvement of claim 29, further comprising
a process for computer visualization of the neurons on a computer screen that
shows one or more neuronal parameters for each neuron according to its
associated location.

  11. 11.

Claim 31 (currently amended): The improvement of claim 21, further comprising
a process for updating the neural network recursively from each indexed time
$t$ to next indexed time $t+1$
wherein the said process recursively decomposes the computation of an
unbounded number of consecutive life times optimality in maximum-likelihood
from an initial time to time $t+1$ into:
using a previous neural network machine that has already optimally computed
recursively from the initial time to time $t$, and determining a computation
of an optimal neural network machine for time $t+1$; and
wherein the said recursive decomposition is conditioned on an incremental
learning restriction, a sensory experience, and a limited resource.

  12. 12.

Claim 32 (currently amended): The improvement of Claim 21 further comprising a
process for incrementally generating neurons through an experience of training
or exploration.

  13. 13.

Claim 33 (currently amended) The improvement of claim 21, further comprising:
an initialization process for constructing the neural network that specifies
whether to connect with one or more of the X, Y, and Z areas; and
a synaptic maintenance process for changing the fluid hierarchy;
wherein each neuron is classified as pre-synaptic, post-synaptic, or both and
during the synaptic maintenance, each post-synaptic neuron determines whether
to cut an input connection where a deviation of weights is high and to re-
connect an input connection when the deviation of weights is low.

  14. 14.

Claim 34 (previously amended): The improvement of claim 21, further comprising
a learning process, wherein the neural network learns an emergent Turing
machine or an emergent universal Turing machine as representations of
experience of teaching or exploration.

  15. 15.

Claim 35 (previously amended): The improvement of claim 21, wherein the fluid
hierarchy internally spans a lifetime.

  16. 16.

Claim 36 (previously amended): The improvement of claim 21, wherein the neural
network selectively uses or selectively disregards combinations of features at
different levels of representations within the fluid hierarchy.

  17. 17.

Claim 37 (currently amended): The improvement of claim 21,
wherein the Z area of the neural network associated with motor information has
at least one overt and at least one covert neuron for each of a plurality of
effectors;
wherein the neural network is trained with one or more plans with or without
task costs using the explicitly taught overt and covert neurons; and
wherein a recalled experienced skill that is not exactly the same as taught or
explored is called an emergent skill.

  18. 18.

Claim 38 (currently amended): The improvement of claim 37, where the neural
network chooses to use one plan from the one or more plans after the network
has been trained with one or more plans.

  19. 19.

Claim 39 (previously amended): The improvement of claim 21, further comprising
a use of the neural network for at least two modalities of vision, audition,
natural language, in a real or a simulated environment.

  20. 20.

Claim 40 (previously amended): The improvement of claim 21, further comprising
a use of the neural network for strong Artificial Intelligence (AI) wherein
the strong AI is not task-specific but is conditioned on incremental learning
framework restrictions, a learning or exploration experience, and a limited
amount of computational resources.

## ABSTRACT OF THE DISCLOSURE

This invention includes a new type of neural network that is able to
automatically and incrementally generate an internal hierarchy without a need
to handcraft a static hierarchy of network areas and a static number of levels
and the static number of neurons in each network area or level. This
capability is achieved by enabling each neuron to have its own dynamic
inhibitory zone using neuron-specific inhibitory connections.