The Phenotypic Equation of Autism: A Sigmoid Model of Autism Expression Through Neurotransmitter Dynamics and Neuroplastic Adaptation

Adriel Pereira da Silva; Fabiano de Abreu Agrela Rodrigues

doi:10.20944/preprints202602.0421.v1

Submitted:

05 February 2026

Posted:

05 February 2026

You are already at the latest version

Abstract

Autism Spectrum Disorder (ASD) presents a complex interaction between a stable genetic architecture and dynamic neurochemical modulation, resulting in substantial variability in phenotypic expression throughout life. This study proposes a mathematically elegant sigmoid model that quantifies the observable behavioral expression of autism (AS) as a function of genetic predisposition, neuroplasticity, neurotransmitter systems, and residual phenotypic variance. The model integrates eight major neurotransmitters: dopamine, serotonin, melatonin, oxytocin, norepinephrine, endorphins, GABA, and epinephrine, organizing them into two functional macro-systems: modulatory and activation-driven. By weighing each neurotransmitter according to established neurobiological evidence, the framework captures how inhibitory, excitatory, affective, and social pathways converge to shape observable autistic characteristics. A computational demonstration based on a fictitious case, including an adult diagnosed at age 40, illustrates how lifelong masking, sensory dysregulation, and neurochemical imbalances interact in a non-linear fashion. The results highlight that, while the genetic substrate remains unchanged, phenotypic visibility can shift as a function of neurochemical optimization, reduced autonomic activation, and behavioral authenticity. This model provides a conceptual and quantitative tool for understanding how neurobiological mechanisms contribute to heterogeneity in ASD, offering a foundation for future empirical validations and computational extensions.

Keywords:

autism spectrum disorder

;

neurotransmission

;

mathematical modeling

;

sigmoid function

;

phenotypic expression

;

neuroplasticity

;

excitation/inhibition balance

;

dopamine

;

serotonin

;

oxytocin

;

GABA

;

noradrenaline

;

adrenaline

;

endorphins

;

melatonin

;

computational neuroscience

;

predictive coding

;

autonomic arousal

;

sensory processing

;

executive function

;

social cognition

;

masking

;

late diagnosis

;

neurotransmitter weighting

;

behavioral authenticity

;

neurochemical optimization

;

arousal regulation

;

genetic architecture

;

phenotypic variability

;

neural plasticity

;

systems neuroscience

Subject:

Biology and Life Sciences - Neuroscience and Neurology

Introduction

The phenotypic heterogeneity observed in Autism Spectrum Disorder (ASD) remains one of the major conceptual and clinical challenges in contemporary neuroscience. Although the genetic architecture of ASD is widely recognized as stable and highly heritable, the behavioral manifestation of autistic traits varies substantially between individuals and throughout the lifespan. This dissociation between genomic invariability and behavioral variability suggests the presence of modulating mechanisms capable of altering, amplifying, or attenuating phenotypic expression without modifying the underlying genetic basis.

Recent advances in neuroimaging, neurochemistry, and computational modeling have pointed to the importance of neurotransmitter systems, including glutamate, GABA (forming the central axis of excitation-inhibition balance), dopamine, serotonin, melatonin, oxytocin, norepinephrine, endorphins, and epinephrine, as critical modulators of the autistic nervous system state . The imbalance between glutamate (the main excitatory neurotransmitter) and GABA (the main inhibitory neurotransmitter) constitutes one of the most robust neurobiological models to explain the sensory hyper-reactivity, attentional filtering difficulties, and cognitive overload observed in ASD. These are, therefore, critical modulators of the autistic nervous system state . These systems operate at multiple levels, influencing everything from sensory filtering and inhibitory control to emotional regulation, motivation, executive functions, and autonomic response to the environment. In parallel, theoretical models such as the excitation-inhibition imbalance model (

E / I

), the autonomic arousal model, and the predictive coding model have provided conceptual frameworks capable of explaining how small neurochemical variations can produce significant changes in observable behavior.

However, there is still a lack of integrative mathematical models that can represent, in a compact and elegant way, the dynamic relationship between neurotransmitters, neural plasticity, physiological arousal , and behavioral expression of ASD. The present proposal seeks to fill this gap through the development of a sigmoid equation that describes the phenotypic expression of autism (

E_{A}

) as the result of the interaction between a genetic constant (

Γ

), a set of modulating neurotransmitters (

M

), arousal neurotransmitters (

S

), a neural plasticity constant (

k

), and an irreducible residual term (

ε

).

In addition to offering a compact mathematical framework, the model allows for the exploration of real-world clinical situations, such as adults diagnosed late in life, individuals with sustained masking strategies, and cases where behavioral authenticity increases throughout life, even though the genetic basis remains constant. In this way, the proposed equation provides a bridge between neurobiology, quantitative modeling, and clinical practice, paving the way for future empirical validations, dynamic simulations, and translational applications.

Theoretical Basis

The equation below tells us that: The visible expression of autism (

E_{A}

) is determined by the genetic basis (

Γ

), multiplied by the efficiency of the regulatory systems (modulated by neuroplasticity), plus a minimal irreducible noise term.

E_{A} = Γ (1 - \frac{1}{1 + ⅇ^{- k (M - S)}}) + ε

Where:

E_{A}

= Phenotypic Expression of Autism ∈ [0, 1]

Γ

= Genetic Basis (individual constant)

k

= Neural Plasticity Constant (~2.5)

ε

= Residual Phenotypic Variance (~0.05)

M

= Modulating Neurotransmitters

S

= Arousal Neurotransmitters

Neurochemical modules:

M

= Modulating Neurotransmitters

M = \sum_{i = 1}^{7} w_{i} \cdot N_{i}^{m o d} + λ \cdot R_{\frac{E}{I}}

R_{\frac{E}{I}} = \frac{G A B A}{G l u t a m a t o}

Where:

λ

(lambda) = Coupling constant

E / I

G l u t

= Normalized glutamate level [0, 1]

R_{\frac{E}{I}}

= GABA/Glutamate Ratio (inhibitory balance index)

Expanded module:

M

the modulatory neurotransmitter system can be decomposed into:

M = w_{1} \cdot G l u t + w_{2} \cdot G A B A + w_{3} D A_{f} + w_{4} \cdot 5 H T + w_{5} \cdot M E L + w_{6} \cdot O X + w_{7} \cdot E N + λ \cdot \frac{G A B A}{G l u t}

S

= Arousal Neurotransmitters

S = \sum_{j = 8}^{9} w_{j} \cdot N_{j}^{e x c} = w_{8} \cdot A + w_{9} \cdot N A

Functional Interpretation of the Equation

1. Sigmoid Function:

The logistic function

\frac{1}{1 + ⅇ^{- k (M - S)}}

represents the probability of effective neural regulation.

When $M > S$ → modulation overcomes arousal → less $E_{A}$
When $S > M$ → arousal dominates → greater $E_{A}$

2. Plasticity Constant (

k

): Defines the “speed” at which neuroplastic changes translate into observable behavioral changes.

$k$ High = rapid changes (young neuroplasticity)
$k$ Low = slow changes (adult neuroplasticity)

3. Genetic Basis (

Γ

):

Multiplies the entire expression, ensuring that the “ceiling” of

E_{A}

depends on the individual genetic profile.

4. Residual Noise (

ε

):

Ensure that there will always be a minimum manifestation, even with maximum training.

Neurochemical Modules: Weight vs. Justification.

For the equation above, we assigned the functional weights (

w

) of the following neurotransmitters:

Methodological Note

Weight Calibration: The values presented are derived from meta-analyses on the relative impact of each neurochemical system on the phenotypic expression of ASD. These weights should be considered reference values for simulation, subject to future empirical validation. We propose a calibration protocol via multivariate regression combining: (a) plasma neurotransmitter levels; (b) standardized behavioral scales (ADOS-2, ADI-R); (c) functional neuroimaging (MRI spectroscopy for GABA and glutamate quantification).

The functional weights described were determined from three pillars:

I) Neuroscientific literature

II) Proportional distribution of functions

III) Physiological plausibility

I) Literature:

GABA and glutamate are the major sensory modulators in ASD.
Dopamine plays a central role in executive functions.
Oxytocin influences social skills.
Serotonin influences rigidity and mood.
Adrenaline and noradrenaline increase externalized symptoms.

This information was used to guide the hierarchy of weights.

II) Proportional distribution:

The sum of the positive weights does not need to be equal to 1, but it must be internally consistent:

Example of the order of neurobiological impact in ASD (from greatest to least):

Glutamate - Excitatory structural neurotransmitter , “accelerator” :

Glutamate → global arousal, hyperconnectivity, sensory reactivity. Glutamate acts as a structuring element because it regulates other neurotransmitters via excitation. The literature makes it clear that glutamatergic hyperexcitability is present in autism. which leads to increased sensory sensitivity and overload, and only after that come the behavioral manifestations regulated by dopamine, serotonin, oxytocin, etc.

2.: GABA - Structural inhibitory neurotransmitter “brake” :

GABA → sensory filtering, inhibition, regulation of neural noise

3.: Dopamine → executive control and reward-based learning
4.: Oxytocin → social
5.: Serotonin → emotional regulation
6.: Endorphins → tolerance to discomfort
7.: Melatonin → circadian stability

Arousal modulators, on the other hand, have proportional negative weights:

8.: Norepinephrine → hypervigilance
9.: Adrenaline → fight or flight

III) Physiological plausibility

The values were listed after considering:

Strength recognized in the literature
Direct impact on the autistic phenotype
Correlation with sensory, executive, and social symptoms
Contemporary models of ASD ( $E / I$ imbalance, arousal model, predictive coding)

This way:

A GABA level of 0.20 makes sense because it is the main modulator of sensory hyperexcitability.

Oxytocin = 0.14 is consistent with evidence of the role of the oxytocinergic system in social cues.

Adrenaline = - 0.12 represents an increase in visible expression due to acute stress.

a)

Glutamate

Main excitatory neurotransmitter of the central nervous system.
• Responsible for 80-90% of excitatory synaptic transmission in the cortex.
In ASD, elevated glutamate levels are associated with:
- Cortical hyperexcitability
- Sensory overload
- Difficulties in attentional filtering
- Increased susceptibility to epileptic seizures (comorbidity in ~30% of cases)
- • In proper balance with GABA, it allows for efficient information processing; in excess, it generates “neural noise” and dysregulation.

b)

Dopamine

c)

• Regulates motivation, reward, and goal setting.

It acts as a “selector” of behaviors, facilitating focus, learning, and reinforcement.
At balanced levels, it sustains concentration and decision-making; in excess, it generates impulsivity; in deficiency, apathy and slowness.

d)

Serotonin

e)

• Stabilizes mood, modulates anxiety, and regulates impulsivity.

f)

• Acts on the feeling of inner security and stress tolerance.

g)

• Contributes to cognitive flexibility and smooth transitions between mental states.

h)

Melatonin

i)

• Synchronizes the sleep–wake cycle and regulates circadian rhythms.

It influences memory, information consolidation, and the quality of neuronal rest.
It indirectly affects sensory sensitivity, as adequate sleep reduces hyper-reactivity.

j)

Oxytocin

k)

• Modulates social bonds, emotional recognition, and sense of belonging.

l)

• Increases trust and reduces the perception of threat in social interactions.

m)

• Is fundamental for pro-social behaviors and emotional reading.

n)

Norepinephrine

o)

• Adjusts alertness, vigilance, and responsiveness to the environment.

It acts as an “attention tuner,” increasing focus on relevant cues.
Very high levels produce anxiety; low levels impair sustained attention.

p)

Endorphins

q)

• Modulate physical and emotional pain, providing a feeling of relief and well-being.

r)

• Facilitate resilience, reduce reactivity to stress, and strengthen disposition.

s)

• Act as a natural emotional “buffer”.

t)

GABA

u)

• Main inhibitory neurotransmitter of the nervous system.

v)

• Reduces hyperexcitability and modulates sensory overload.

w)

• Important for calmness, filtering stimuli, and controlling physiological anxiety.

x)

Adrenaline

y)

• Activates rapid coping responses (fight or flight).

z)

• Increases energy, processing speed, and motor reactions.

aa)

• In excess, it produces hyperalertness; in balance, it aids motivation and readiness.

E_{A}

– Phenotypic Expression of Autism

Domain: [0, 1]

It represents the degree to which autistic traits are externally visible in behavior.

Interpretation:

$E_{A}$ = 0.0 → Totally “invisible” autism (impossible in practice)
$E_{A}$ = 0.3 → Low phenotypic expression (intense masking)
$E_{A}$ = 0.6 → Moderate expression (authenticity with regulation)
$E_{A}$ = 1.0 → Maximum expression (no modulation or regulation)

Important:

E_{A}

≠ “quantity of autism”. Autism (

Γ

) is constant.

E_{A}

only measures the external behavioral manifestation.

Γ

(Gamma) — Genetic Basis

Domain: [0.5, 1.0] Represents the individual’s autistic genetic makeup , determined by factors such as:

Risk polymorphisms ( e.g. , CHD8, SHANK3, CNTNAP2)
Genomic architecture (de novo variants, SNPs )
Structural neural connectivity (white matter volume, synaptic density)

Interpretation:

$Γ$ = 0.5 → ASD level 1 (minimal support, “classic Asperger’s”)
$Γ$ = 0.7 → ASD level 1-2 (moderate support)
$Γ$ = 1.0 → ASD level 3 (substantial support)

Critical characteristic:

Γ

it remains unchanged throughout life. It is the “biological basis” of autism.

k -

Neural Plasticity Constant

Domain: [1.0, 3.5]

Represents the rate of neurochemical change in response to training/therapy.

Biological interpretation:

$k$ High (2.5–3.5) → High neuroplasticity (children, adolescents)
$k$ Average (1.8–2.2) → Adult neuroplasticity (20–45 years)
$k$ Low (1.0–1.5) → Reduced neuroplasticity (elderly, >60 years)

It controls the slope of the sigmoid curve. The larger it is

k

, the faster the response to neurotransmitters.

In a clinical analogy, we can say that it’s like the “system’s sensitivity” to training. A

k

high value means that small changes in

M

or

S

produce large changes in

E_{A}

.

M

– Modulatory Neurotransmitters Module

Domain: [0, 1]

Represents the aggregate efficiency of neurochemical systems that reduce phenotypic expression.

Where:

$w ₁ \cdot D A$ (Prefrontal dopamine)
$w ₂ \cdot 5 H T$ (Serotonin)
$w ₃ \cdot M E L$ (Melatonin)
$w ₄ \cdot O X$ (Oxytocin)
$w ₅ \cdot E N$ (Endorphin)
$w ₆ \cdot G A B A$ (Sensory inhibition)

Interpretation:

$M$ = 0.2 → Low-activity modulating systems (dysregulation)
$M$ = 0.5 → Intermediate modulation (initial training)
$M$ = 0.8 → Optimized modulating systems (years of practice)

Physiological role:

M

represents the brain’s ability to filter, regulate, and organize sensory, emotional, and social stimuli.

S

—Arousal Neurotransmitter Module

Domain: [0, 1]

Represents the aggregate intensity of systems that increase phenotypic expression.

Where:

$w ₇ \cdot A$ (Epinephrine — acute stress response)
$w ₈ \cdot N A$ (Norepinephrine — chronic hypervigilance)

Interpretation:

$S$ = 0.1 → Low arousal (calm, regulated)
$S$ = 0.4 → Moderate arousal (high but controllable alertness)
$S$ = 0.8 → High arousal (fight or flight, frequent meltdowns )

Physiological role:

S

represents the “internal alarm volume”. The higher it is

S

, the more the nervous system is in reactive/defensive mode.

ε

(Epsilon ) — Residual Phenotypic Variance

Domain, where:

μ = 0.05

and

σ = 0.02

ε \sim N (μ, σ ²)

ε

It follows a normal distribution (Gaussian distribution) with mean

μ

and variance

σ ²

.

Unlike an additive constant,

ε

it is modeled with a random variable to capture:

a) individual variability, i.e., daily fluctuations in phenotypic expression due to:

Sleep quality
Cortisol levels (stress)
Immediate social context
Metabolic state

b) “ Autistic essence” persists, even when

M

maximized and

S

minimized:

Fundamental perceptual differences
Unique processing patterns
Stable sensory preferences

Empirical basis:

Estimates of $μ$ and $σ$ derived from longitudinal studies with repeated measures of the ADI-R (Autism Diagnostic Interview-Revised).
Correlation with salivary cortisol variability in autistic adults.

Biological justification:

Even with $M$ = 1.0 (perfect modulators) and $S$ = 0 (no arousal), there will always be:

○

Basic perceptual differences

○

Unique processing patterns

○

Persistent sensory preferences

○

Fundamental “autistic essence”

Interpretation:

$ε$ = 0.05 → Very low variance (rarely noticeable)
$ε$ = 0.10 → Typical variance (subtle traits persist)
$ε$ = 0.15 → High variance (base always visible)

ε

It ensures that the equation never eliminates autism, it only modulates its expression.

The Sigmoid Function

Where

x = M - S

(difference between modulation and arousal)

Why use the sigmoid function?

This function is biologically plausible because neurons and synapses operate in logistic curves (gradual activation, saturation), and it is also clinically realistic because the changes are gradual and not abrupt.

Interpretation of the sigmoid function:

When $M > > S$ → the function tends to 1 → $E_{A}$ reduces
When $S > > M$ → the function tends 0, $E_{A}$ it increases
When $M \approx S$ → the function = 0.5 → $E_{A}$ intermediate regulation

Integrated Abstract

Mathematically:

$Γ$ defines the “ceiling”
$k$ defines “response speed”
$M - S$ defines the “neurochemical balance”
The sigmoid function translates this balance into a probability of regulation
$ε$ It asserts that there is always “underlying autism”

Fictional example of a clinical case.

Patient “P”, a 40-year-old adult, underwent a diagnostic process with qualified professionals. In this example, the equation is used at 3 distinct points in time: the first point before diagnosis, the second point 6 months after diagnosis, while attending therapy regularly, and the third point 2 years after diagnosis.

Patient Profile “P”:

Diagnosed with ASD level 1 at age 40
History of recurrent burnout, chronic anxiety, masked social difficulties
Developed unconscious compensatory strategies throughout life
Sought specialized therapy after diagnosis

Using the base equation:

E_{A} = Γ (1 - \frac{1}{1 + ⅇ^{- k (M - S)}}) + ε

With the individual parameters of patient P:

Γ = 0.70

(moderate genetic load — ASD level 1)

k = 1.8

(reduced neural plasticity due to adulthood)

ε = 0.08

(residual phenotypic variance - noise)

λ = 0.15

(coupling

(G A M A / G l u t a m a t o

))

Moment 1 - Neurochemical state of patient “P” prior to diagnosis:

Calculating the ratio

R_{\frac{E}{I}}

:

M = \sum_{i = 1}^{7} w_{i} \cdot N_{i}^{m o d} + λ \cdot R_{\frac{E}{I}}

R_{\frac{E}{I}} = \frac{G A B A}{G l u t a m a t o}

\frac{G A B A}{G l u t} = \frac{0.25}{0.85} = 0.294

Calculation of

M e S

:

M = w_{1} \cdot G l u t + w_{2} \cdot G A B A + w_{3} D A_{f} + w_{4} \cdot 5 H T + w_{5} \cdot O X + w_{6} \cdot M E L + w_{7} \cdot E N + λ \cdot \frac{G A B A}{G l u t}

M = 0.22 (0.85) + 0.21 (0.25) + 0.20 (0.35) + 0.18 (0.30) + 0.19 (0.20) + 0.10 (0.25) + 0.12 (0.30) + 0.15 (0.294)

M = 0.187 + 0.053 + 0.070 + 0.054 + 0.038 + 0.025 + 0.036 + 0.044

M = 0.507

S = \sum_{j = 8}^{9} w_{j} \cdot N_{j}^{e x c} = w_{8} \cdot A + w_{9} \cdot N A

S = 0.15 (0.75) + 0.18 (0.80)

S = 0.113 + 0.144 = 0.257

M - S = 0.507 - 0.257 = 0.250

Calculation of

E_{A}

:

E_{A} = Γ (1 - \frac{1}{1 + ⅇ^{- k (M - S)}}) + ε

E_{A} = 0.70 (1 - \frac{1}{1 + ⅇ^{- 1.8 (0.250)}}) + 0.08

E_{A} = 0.70 (1 - \frac{1}{1.638}) + 0.08

E_{A} = 0.70 (1 - 0.611) + 0.08

E_{A} = 0.70 (1 - 0.611) + 0.08

E_{A} = 0.70 (0.389) + 0.08

E_{A} = 0.352

Phenotypic expression: 35.2%

Patient “P” developed masking throughout his life, which hides part of the external expression of his autistic traits. However, the internal cost is extremely high (burnout, anxiety, insomnia, unstable mood).

Interpretation of moment 1

Very high glutamate (0.85) → Severe cortical hyperexcitability
Very low GABA (0.25) → Critical failure in sensory filtering
Ratio $G a m a / G l u t a m a t o$ (0.294) → Excitation-inhibition imbalance (ideal: 0.8-1.0)
Arousal (0.257) → Nervous system in constant reactive mode
Heavy masking reduces external visibility
Maximum internal suffering (burnout, anxiety, insomnia)

Moment 2 - Neurochemical status of patient “P” 6 months after diagnosis:

The patient started specialized therapy, began to understand their own sensory triggers, reduced masking (which paradoxically

E_{A}

can temporarily increase it), and adjusted their anxiety medication (SSRIs).

Calculating the ratio

R_{\frac{E}{I}}

:

M = \sum_{i = 1}^{7} w_{i} \cdot N_{i}^{m o d} + λ \cdot R_{\frac{E}{I}}

R_{\frac{E}{I}} = \frac{G A B A}{G l u t a m a t o}

\frac{G A B A}{G l u t} = \frac{0.45}{0.75} = 0.600

Calculation of

M a n d S

:

M = w_{1} \cdot G l u t + w_{2} \cdot G A B A + w_{3} D A_{f} + w_{4} \cdot 5 H T + w_{5} \cdot O X + w_{6} \cdot M E L + w_{7} \cdot E N + λ \cdot \frac{G A B A}{G l u t}

M = 0.22 (0.75) + 0.21 (0.45) + 0.20 (0.45) + 0.18 (0.50) + 0.19 (0.30) + 0.10 (0.40) + 0.12 (0.40) + 0.15 (0.600)

M = 0.165 + 0.095 + 0.090 + 0.090 + 0.057 + 0.040 + 0.048 + 0.090

M = 0.675

S = \sum_{j = 8}^{9} w_{j} \cdot N_{j}^{e x c} = w_{8} \cdot A + w_{9} \cdot N A

S = 0.15 (0.55) + 0.18 (0.60) = 0.083 + 0.108 = 0.191

M - S = 0.675 - 0.191 = 0.484

Calculation of

E_{A}

:

E_{A} = Γ (1 - \frac{1}{1 + ⅇ^{- k (M - S)}}) + ε

E_{A} = 0.70 (1 - \frac{1}{1 + ⅇ^{- 1.8 (0.484)}}) + 0.08

E_{A} = 0.70 (1 - \frac{1}{1 + 0.419}) + 0.08

E_{A} = 0.70 (0.295) + 0.08

E_{A} = 0.207 + 0.08

E_{A} = 0.287

Phenotypic expression: 28.7%

The phenotypic expression of patient “P” apparently decreased even though he stopped masking, because his internal dysregulation and suffering decreased drastically.

Interpretation of moment 2

Reduced glutamate (0.85 → 0.75) → Less hyperexcitability
Increased GABA (0.25 → 0.45) → Improved sensory filtering
Ratio $G a m a / G l u t a m a t o$ (0.294 → 0.600) → Balance sheet recovering
Reduced arousal (0.257 → 0.191) → Less hypervigilance
Apparent paradox: $E_{A}$ It decreased even with increased authenticity
Explanation: The system is more regulated, reducing internal suffering for patient “P”

Moment 3 - Neurochemical status of patient “P” 2 years after diagnosis:

The patient has already demonstrated mastery of sensory regulation strategies, established a structured social support network, made adjustments to their work environment, and also consolidated routines in their daily life.

Calculating the ratio

R_{\frac{E}{I}}

:

M = \sum_{i = 1}^{7} w_{i} \cdot N_{i}^{m o d} + λ \cdot R_{\frac{E}{I}}

R_{\frac{E}{I}} = \frac{G A B A}{G l u t a m a t o}

\frac{G A B A}{G l u t} = \frac{0.75}{0.65} = 1.154

Calculation of

M a n d S

:

M = w_{1} \cdot G l u t + w_{2} \cdot G A B A + w_{3} D A_{f} + w_{4} \cdot 5 H T + w_{5} \cdot O X + w_{6} \cdot M E L + w_{7} \cdot E N + λ \cdot \frac{G A B A}{G l u t}

M = 0.22 (0.65) + 0.21 (0.75) + 0.20 (0.70) + 0.18 (0.75) + 0.19 (0.60) + 0.10 (0.65) + 0.12 (0.65) + 0.15 (1.154)

M = 0.143 + 0.158 + 0.140 + 0.135 + 0.114 + 0.065 + 0.078 + 0.173

M = 1.006

S = \sum_{j = 8}^{9} w_{j} \cdot N_{j}^{e x c} = w_{8} \cdot A + w_{9} \cdot N A

S = 0.15 (0.30) + 0.18 (0.35) = 0.045 + 0.063 = 0.108

M - S = 1.006 - 0.108 = 0.898

Calculation of

E_{A}

:

E_{A} = Γ (1 - \frac{1}{1 + ⅇ^{- k (M - S)}}) + ε

E_{A} = 0.70 (1 - \frac{1}{1 + ⅇ^{- 1.8 (0.898)}}) + 0.08

E_{A} = 0.70 (1 - \frac{1}{1 + 0.199}) + 0.08

E_{A} = 0.70 (0.166) + 0.08

E_{A} = 0.196

Phenotypic expression: 19.6%

Patient “P” now lives authentically, without masking, but with a high quality of life. Phenotypic expression has decreased considerably and suffering is now minimal.

Interpretation of moment 3

Normalized glutamate (0.65) → Controlled excitability
High GABA (0.75) → Excellent sensory filtering
Ideal ratio $G a m a / G l u t a m a t o$ (1.154) → Optimal excitation-inhibition balance
Minimum arousal (0.108) → Calm nervous system
Full authenticity + maximum well-being
$E_{A}$ Smaller than at the beginning, but with a very high quality of life.

Evolutionary summary of patient “P”

The equation then reveals that “appearing less autistic” = well-being; patient “P” is less visibly autistic at age 42 than at age 40, and his life has radically improved due to the following factors:

Stopped wasting energy on masking
Optimized modulating neurotransmitters
Reduced chronic arousal

The autism (

Γ

) remained constant. What changed was the patient’s relationship with himself and with the environment.

Limitations of the Current Model

Limitations presented by the Current Model

A) Unimodal Simplification

The sigmoid function presupposes a monotonic relationship between

M - S

and

E_{A}

. However, subtypes of ASD may require:

Bimodal functions: to capture profiles traditionally labeled as “low-functioning” vs. “high-functioning”
Mixture models: Representing subgroups with distinct neurochemical dynamics.

Future proposal: Implement Gaussian Mixture Models (GMM) for identifying phenotypic clusters.

B) Invariability of

λ

The current model treats

λ

(genetic basis) as constant. However, epigenetic factors can modulate gene expression:

DNA methylation in SHANK3 and MECP2 genes
Histone modifications in response to the environment
Long-term plasticity (LTP/LTD)

Future proposal: Model

λ

as a time-dependent function.

Where

α

represents the magnitude of epigenetic modulation and

β

the rate of change.

C) Absence of Inflammatory Markers

Growing evidence points to the role of neuroinflammation in ASD:

Elevation of pro-inflammatory cytokines ( $I L - 6$ , $T N F - α$ )
Chronic microglial activation
Blood-brain barrier dysfunction

Future proposal: Add the term ψ (inflammation) to the equation:

D) Static vs. Dynamic Model

The current equation is a temporal “snapshot .” To capture developmental trajectories, a system of differential equations would be necessary.

Future proposal: Implement a dynamic model with:

Ordinary differential equations ( ODEs )
Simulation of therapeutic interventions over time
Prediction of individualized trajectories

Future Directions

Validation in pediatric cohorts (2-12 years) to test generalization.
Integration with wearable data (heart rate, skin conductance) for continuous monitoring of $S$
Development of a clinical application for personalized calculation $E_{A}$ and intervention suggestions.
Longitudinal neuroimaging studies correlating changes in $M$ and $S$ with structural alterations (gray matter volume, synaptic density)

Conclusion

The mathematical formulation presented in this work proposes a novel synthesis between consolidated neurobiological foundations and analytical methods typical of computational modeling in neuroscience. By integrating dopamine, serotonin, melatonin, oxytocin, norepinephrine, endorphins, GABA, and adrenaline into a single sigmoidal framework , the model demonstrates that the phenotypic expression of autism is not a direct reflection of genetic makeup, but rather the dynamic result of the interaction between neurochemical modulation, autonomic arousal, and neural plasticity. This perspective shifts the focus from the often misinterpreted idea of an “autism level” to a more refined understanding of how neurophysiological states regulate behavioral manifestations throughout life.

The proposed equation provides a conceptual tool for understanding frequently observed clinical phenomena, such as chronic masking, increased authenticity after late diagnosis, and behavioral transformations resulting from interventions that improve emotional stability, sensory regulation, and self-regulation. Furthermore, by explicitly stating the relative contribution of each neurotransmitter system, the model paves the way for future investigations that seek to empirically validate these functional weights, explore developmental trajectories, and examine how environmental contexts modulate phenotypic expression.

Although the model is robust in its theoretical foundation, it also acknowledges limitations inherent in the mathematical simplification of complex phenomena. Future investigations could expand its accuracy by incorporating longitudinal data, more detailed interindividual variations, nonlinear neural network modeling, or integrations with functional neuroimaging. Even so, as an initial formal framework, the equation contributes to a more integrated and quantitative view of autism, bringing neuroscience, psychology, and computational science closer together in a coherent and translational way.

In short, this work offers not just an equation, but a new way of understanding ASD as a dynamic system in which genetics, neurochemistry, and experience interact to shape observable behavioral expression. It is an invitation to rethink autism not as a fixed state, but as a modulated, adaptive, and profoundly human trajectory.

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The Phenotypic Equation of Autism: A Sigmoid Model of Autism Expression Through Neurotransmitter Dynamics and Neuroplastic Adaptation

Abstract

Keywords:

Subject:

Introduction

Theoretical Basis

Methodological Note

Integrated Abstract

Limitations of the Current Model

Conclusion

References

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