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A Boltzmann machine is a stochastic neural network that learns a probability distribution specified by pairwise energies. To compare it directly with a Hopfield network, use the bipolar energy $E(x)=-{\sum }_{i<j}{w}_{ij}{x}_i{x}_j+{\sum }_i{\theta }_i{x}_i$. Here $\beta >0$ controls the preference for low energy, and the partition function $Z={\sum }_x{e}^{-\beta E(x)}$ normalizes all states. In a computational model, $\beta$ is an algorithmic inverse temperature; interpreting it as $1/(k_{B}T)$ additionally requires physical energy units.

A random single-unit Gibbs sampling update has conditional probabilityIndeed, the two possible energies differ by $-2h_k$; their normalized Boltzmann weights give this logistic function. Random selection of one unit at a time satisfies detailed balance. For finite weights and finite positive $\beta$, all conditional probabilities lie strictly between zero and one, so the finite Markov chain can reach every state and can remain in place. Its unique stationary distribution is $p_{\beta }$. A short run need not yet sample that distribution accurately.

As $\beta \to \infty$, a nonzero field selects its sign, approaching asynchronous Hopfield energy descent. At zero field the stochastic rule remains a fair coin, whereas the deterministic convention above retains the old value. At finite inverse temperature, uphill moves permit exploration beyond one local minimum. Recall and learning are different operations: sampling changes the units at fixed weights; learning changes the weights to fit the training data.

The Bradley-Terry model assigns positive abilities ${\theta }_i$ and the displayed pairwise winning probabilities. Multiplying all abilities by the same positive scalar leaves the probabilities unchanged, so identifiability requires a normalization. In log abilities ${\beta }_i=\log {\theta }_i$, the model has comparison probabilities obtained from the logistic function. Either parametrization induces the same ordering of players.

Orient the edges of a comparison tree and set ${\eta }_{ij}=\log {\theta }_i-\log {\theta }_j$. After anchoring one vertex strength, these edge quantities are unconstrained coordinates. The Bradley-Terry model probability is the logistic function of ${\eta }_{ij}$, and its binomial log-likelihood contribution is $m[p{\eta }_{ij}-\log (1+e^{{\eta }_{ij}})]$. Differentiating gives the fitted odds $p/(1-p)$.

The logistic differential equation is the nonlinear ordinary differential equationAs a mathematical model of population dynamics, it treats abundance $N$ as a continuous nonnegative quantity. The constant $r$ has units of inverse time, while the carrying capacity $K$ has the same units as $N$. The useful idea is to combine growth resembling exponential growth at low abundance with a slowing approach to a finite abundance. The three trajectories below already show an important distinction: starting above the carrying capacity gives a decreasing curve, whereas starting below it gives an increasing one.

For $N>0$, dividing the derivative by abundance gives the instantaneous per-capita rate:It decreases linearly with $N$: it approaches $r$ at low abundance, is $r/2$ at $N=K/2$, vanishes at the carrying capacity, and is negative above it. This per-capita rate differs from the total rate $N^{\prime }$, which also counts how many individuals contribute. It is an instantaneous rate, rather than a fixed percentage increase over a whole day. The linear per-capita assumption and the resulting qualitative picture are developed in MIT 18.03SC's logistic-model notes.

Writing $f(N)=rN(1-N/K)$, the equilibrium points satisfy $f(N)=0$, so the constant solutions of a differential equation are $N(t)=0$ and $N(t)=K$. The phase line points upward for $0<N<K$ and downward for $N>K$. Also $f^{\prime }(0)=r>0$ and $f^{\prime }(K)=-r<0$: the criterion for fixed point stability for an autonomous differential equation makes zero an unstable equilibrium and $K$ an asymptotically stable equilibrium point. Every positive initial condition approaches $K$ as $t\to \infty$; $N_0=0$ stays exactly zero. Instability of zero means that a small positive population grows away from it, not that a zero population spontaneously appears.

For every $N_0>0$, the unique solution of a differential equation for all $t\ge 0$ isThe denominator is positive on this domain, including when $N_0>K$. For $0<N_0<K$, this is a translated and scaled logistic function; its extension to all real times is a sigmoid function. For $N_0>K$ the coefficient of the exponential function is negative, and the future curve instead decreases toward $K$. No nonconstant positive solution of a differential equation crosses $K$ or reaches it at finite time. A short derivation uses the dimensionless variable $z=K/N-1$: the chain rule gives $z^{\prime }=-rz$, so exponential decay yields $z(t)=(K/N_0-1)e^{-rt}$. The substitution divides by $N$, so the zero equilibrium point must be included separately.

The total growth rate is a downward-opening quadratic function of abundance. Completing the square gives its maximum without solving the differential equation:The balance is beautiful: more individuals contribute to growth, but each contributes less, and the strongest total increase occurs halfway to the carrying capacity. This is a maximum over possible abundances. A trajectory starting above $K/2$ never passes through that abundance in its future.

The chain rule connects this maximum to the shape of the time trajectory through its second derivative:For $0<N_0<K/2$, the trajectory initially has convexity and then concavity, with its one future inflection point atFor $N_0=K/2$, this point lies at the initial time, with concavity for $t>0$. For $K/2<N_0<K$, the inflection point belongs to the past extension, so the future trajectory has concavity throughout. Above $K$, the decreasing trajectory has convexity and no inflection point. The two constant solutions of a differential equation have no change of concavity. Thus an entire logistic function has an S shape, but its observed future segment need not show both bends. See also ^[ref].

For the plotted example with $K=1000$, $r=0.4{\mathrm{day}}^{-1}$ and $N_0=100$, time $t$ measured in days givesThe initial total rate is $N^{\prime }(0)=0.4\cdot 100\cdot 0.9=36$ individuals per day, and the initial per-capita rate is $0.36{\mathrm{day}}^{-1}$. The fastest total increase is $rK/4=100$ individuals per day, reached at $t_{1/2}=\ln 9/0.4\approx 5.49306$ days. At that time the per-capita rate has already fallen to $0.2{\mathrm{day}}^{-1}$.

These parameters are illustrative choices, rather than an empirical fit. The equation specifies deterministic evolution under a phenomenological assumption: it summarizes how net growth depends on abundance without separately modelling every process that affects it. The carrying capacity is a parameter for a specified setting, not a universal constant for all populations. In his 1838 paper "Notice sur la loi que la population suit dans son accroissement", Correspondance mathématique et physique 10, pp. 113–121, Pierre-François Verhulst proposed $p^{\prime }=mp-n{p}^2$, equivalent to this equation with $r=m$ and $K=m/n$. He also noted that different assumed growth-retarding terms could describe the observations similarly while giving very different limiting populations. This is an early illustration of why a good-looking curve alone does not establish a unique long-term prediction.^[ref]

The logistic differential equation describes continuous time. The logistic map instead advances in discrete steps and can exhibit a period-doubling cascade and Devaney chaos; the monotone positive trajectories here cannot. Their connection through the forward Euler method is explained under Euler discretization of logistic growth, where a large step can introduce behaviour absent from the original differential equation.

The logistic growth equation applies the logistic differential equation to population dynamics. For positive growth rate $r$ and carrying capacity $K$, an initial condition $0<N(0)<K$ gives $N(t)=K/[1+b{e}^{-rt}]$, where $b=K/N(0)-1$. Its per-capita growth falls linearly with abundance, and a trajectory starting below $K/2$ reaches an inflection point when $N=K/2$. The exact solutions of a differential equation and their diagrams are developed under logistic differential equation. A translated and scaled logistic function has the same form. In nonlinear regression for biological size, an analogous saturation curve need not imply that circumference itself obeys a mechanistic population law.

Include an intercept $w_0$ and let $\mathbf{z}$ be the chosen numerical feature vector. Logistic regression usesThe logistic function maps every real score into $(0,1)$, allowing a Bernoulli distribution model for the label. Equivalently, the log odds are linear in the features:This gives a smooth differentiable probability model with a linear decision boundary at any fixed probability threshold. The sigmoid output is a fitted conditional probability, not a categorical prediction until a threshold is chosen.

Human disorders show that motivation depends on several distinguishable operations: initiating activity, translating incentive value into effort, evaluating outcomes and sustaining goal-directed behavior. A person may be able to execute an instructed movement yet fail to initiate useful action, or may respond differently to reward and effort without a global loss of intelligence. These dissociations argue against equating motivation with motor strength or with a single quantity of pleasure.

A striking starting point is auto-activation deficit. Dominique Laplane and colleagues described three people with marked reduction in spontaneous mental and behavioral activity after toxic encephalopathy and bilateral basal ganglia brain lesions, especially involving the globus pallidus on the available scans. Intellectual capacities were relatively preserved and motor impairment could be mild. External stimulation could reverse the apparent inactivity. This contrast suggested a problem in self-activation rather than inability to perform every relevant cognitive or motor operation.^[ref]

The cases were mechanistically informative but not anatomically pure experiments. Toxic injury can affect more than the structures visible on a scan, and reduced initiation can interact with executive, emotional and motor processes. The observation therefore implicates basal ganglia–frontal systems in internally generated activity without showing that one nucleus contains all motivation. It also motivates assessing spontaneous behavior separately from performance after repeated prompting.

Liane Schmidt and colleagues tested a more specific incentive-to-action hypothesis. They contrasted an instructed handgrip task, requiring specified force levels, with an incentive task where effort could earn money. Patients with basal ganglia damage and auto-activation difficulty could use external instructions to produce force but showed abnormal translation of incentives into effort. The apparatus makes the dissociation concrete: an available motor response does not ensure that anticipated reward will appropriately energize it.^[ref]

This helps separate motor capacity from motivational use of that capacity. Nevertheless, monetary incentives also require understanding task contingencies, representing value and maintaining the goal. A reduced force response does not uniquely identify absence of pleasure. The strength of the design is the contrasting instruction condition; the remaining question is which valuation, activation or control operation prevents self-driven use of the preserved response.

Parkinson's disease offers converging evidence while introducing its own confounds. Campbell Le Heron and colleagues tested 39 patients, with and without clinical apathy, ON and OFF their usual dopaminergic medication, alongside 32 controls. Choices varied both monetary reward and required physical effort. Apathy and medication state were associated with different patterns across the reward–effort space, rather than one uniform decrease in willingness to act. This supports distinguishing clinical motivational difficulty from the complete set of effects produced by changing dopamine availability.^[ref]

The design is stronger than comparing raw movement speed alone, but an ON/OFF contrast is not a selective perturbation of one psychological computation. Medication changes motor function and potentially learning, alertness and response vigor. Apathy groups are not randomly assigned, and comprehension, depression, fatigue and motor capacity can differ. Interpreting choice patterns therefore requires suitable calibration and comparison measures. Dopamine sensitivity does not justify the statement that dopamine is simply a pleasure signal.

A small statistical model illustrates why effort-based decision-making experiments must vary both benefits and costs. Suppose acceptance depends on $U=\alpha R-\kappa E$, where $R$ is reward, $E$ effort, $\alpha$ reward gain and $\kappa$ effort cost. With a fixed logistic function as the choice rule, a lower reward gain and a higher effort cost can produce exactly the same acceptance probability at one offer. This single-offer reward-effort ambiguity motivates varying both dimensions to distinguish their predictions within the model. These are alternative explanations to test, not parameter estimates obtained merely by diagnosing apathy.

Patients with prefrontal cortex damage reveal another component: evaluation of actions across outcomes and time. Antoine Bechara and colleagues introduced the procedure now called the Iowa gambling task, in which choices offered different short-term gains and longer-term consequences. Patients with ventromedial prefrontal cortex damage continued making disadvantageous choices despite relatively preserved conventional intellectual abilities. This linked real-world decision difficulty to experimentally observable valuation and learning demands.^[ref]

The result supports a role in using consequences to guide behavior, but the task combines reward learning, loss sensitivity, reversal, uncertainty and human memory. It does not uniquely establish one bodily-feedback mechanism or prove that the patients have no emotional responses. Similarly, anhedonia, reduced initiation and willingness to expend effort should be measured separately rather than treated as synonyms. Several mechanisms can produce superficially similar inactivity, and a change in one response measure does not imply a change in every form of motivation.

Across these disorders, the most useful inference is computational and relational. Basal ganglia injury distinguishes instructed action from self-activation and incentive translation; Parkinsonian comparisons distinguish aspects of reward–effort choice from medication-related motor and motivational effects; prefrontal damage reveals difficulty applying future consequences. These operations interact through connected systems and need not map to one region each. Patient studies have transformed motivation from a vague explanation of behavior into separable hypotheses about initiation, valuation, effort and control, while demanding that the preserved capacities and competing explanations be tested explicitly.

Write $A_i={\varphi }_{i1}$, $b_i={\varphi }_{i2}>0$ and $k_i=-{\varphi }_{i3}={\theta }_{i3}$. ThenFor $k_i>0$, $A_i$ is the limiting circumference and $1/(1+b_i)$ is the fraction of that asymptote attained at the centered day $x=\overline{x}$. Under the code's transformations,The latter expression is the logistic function. The curve also satisfies the logistic growth equation $d{m}_i/dx=(k_i/s_x)m_i(1-m_i/A_i)$. Its inflection point is at half the asymptote. The original prior permits $k_i\le 0$, in which case it does not describe increasing growth.

The intercept is the midpoint of the two log odds:Equivalently $e^{\alpha }$ is the geometric mean of the two odds. The logistic function evaluated at $\alpha$ represents a central event probability, and approximates the average of the two group probabilities when $\beta$ is small. It is not generally their arithmetic average, nor is $\alpha$ specifically the control-group log odds.

The Bradley-Terry model assigns comparison probability $q_i={\theta }_i/({\theta }_i+{\theta }_{i+1})$ to each observed edge. Up to factors independent of the parameters, its likelihood function isThe comparison graph is a path graph, hence a tree. After fixing ${\theta }_n=1$, its edge ratios are unconstrained positive coordinates: every collection $r_i={\theta }_i/{\theta }_{i+1}>0$ determines uniquely ${\theta }_i={\prod }_{k=i}^{n-1}{r}_k$.

For a convenient strict-concavity calculation, use the edge log-ratios in a Bradley-Terry comparison tree ${\eta }_i=\log r_i$. The log-likelihood separates asEach derivative is $m(p_i-q_i)$, where $q_i$ is the logistic function of ${\eta }_i$, and each second derivative is $-m{q}_i(1-q_i)<0$. Because $0<p_i<1$, there is a unique finite global maximum atConverting back gives the Bradley-Terry maximum-likelihood estimate on a path:Equivalently, recurse backwards using ${\hat{\theta }}_i={\hat{\theta }}_{i+1}{p}_i/(1-p_i)$. All estimates are finite and positive. The sample size $m$ changes the curvature of the likelihood but not this maximizer; absence of comparison cycles is what permits every empirical edge proportion to be fitted simultaneously.

For $k\in {\mathcal{N}}_i$, put $a_k^{(i)}={\sum }_{j\in {\mathcal{N}}_k\setminus \{i\}}{s}_j$ and $H_i={\sum }_{j\in {\mathcal{N}}_i}{s}_j$. Removing all factors independent of $s_i$ from the posterior distribution gives, for $u\in \{-1,1\}$,A stable implementation uses the difference of the log weights. The conditional log odds areso ${\pi }_i(+1\mid s_{-i},x)=1/(1+e^{-L_i})$, a logistic function. For very large $|L_i|$, evaluate this logistic expression using the sign of $L_i$ to avoid exponential overflow.

For Gibbs sampling for a finite hidden spin field, initialize any spin configuration. At every step choose $i\in D$ uniformly, draw a fresh uniform variable, set spin $i$ to $+1$ with the probability above and to $-1$ otherwise, and leave all remaining spins fixed. This Random-scan Gibbs sampler has the stated posterior distribution invariant by detailed balance of a random-scan Gibbs sampler. All conditional probabilities are strictly between zero and one. For the one-site case, the conditional probabilities are both $1/2$.