Lenia

Let ${\displaystyle {\mathcal {L}}}$ be the lattice or grid containing a set of states ${\displaystyle S^{\mathcal {L}}}$. Like many cellular automata, Lenia is updated iteratively; each output state is a pure function of the previous state, such that

$${\displaystyle \Phi (A^{0})=A^{\Delta t},\Phi (A^{\Delta t})=A^{2\Delta t},\ldots ,\Phi (A^{t})=A^{t+\Delta t},\ldots }$$

where ${\displaystyle A^{0}}$ is the initial state and ${\displaystyle \Phi :S^{\mathcal {L}}\rightarrow S^{\mathcal {L}}}$ is the global rule, representing the application of the local rule over every site ${\displaystyle \mathbf {x} \in {\cal {L}}}$. Thus ${\displaystyle \Phi ^{N}(A^{t})=A^{t+N\Delta t}}$.

If the simulation is advanced by ${\displaystyle \Delta t}$ at each timestep, then the time resolution ${\displaystyle T={\frac {1}{\Delta t}}}$.

Let ${\displaystyle S=\{0,1,\ldots ,P-1,P\}}$ with maximum ${\displaystyle P\in \mathbb {Z} }$. This is the state set of the automaton and characterizes the possible states that may be found at each site. Larger ${\displaystyle P}$ correspond to higher state resolutions in the simulation. Many cellular automata use the lowest possible state resolution, i.e. ${\displaystyle P=1}$. Lenia allows for much higher resolutions. Note that the actual value at each site is not in ${\displaystyle [0,P]}$ but rather an integer multiple of ${\displaystyle \Delta p={\frac {1}{P}}}$; therefore we have ${\displaystyle A^{t}(\mathbf {x} )\in [0,1]}$ for all ${\displaystyle \mathbf {x} \in {\mathcal {L}}}$. For example, given ${\displaystyle P=4}$, ${\displaystyle \mathbf {A} ^{t}(\mathbf {x} )\in [0,0.25,0.75,1]}$.

A 9-square Moore neighborhood like those used in Game of Life.

The "ball" neighborhoods used by Lenia.

Mathematically, neighborhoods like those in Game of Life may be represented using a set of position vectors in ${\displaystyle \mathbb {R} ^{2}}$. For the classic Moore neighborhood used by Game of Life, for instance, ${\displaystyle {\mathcal {N}}=\{-1,0,1\}^{2}}$; i.e. a square of size 3 centered on every site.

In Lenia's case, the neighborhood is instead a ball of radius ${\displaystyle R}$ centered on a site, ${\displaystyle {\mathcal {N}}=\{\mathbf {x} \in {\mathcal {L}}:\lVert \mathbf {x} \rVert _{2}\leq R\}}$, which may include the original site itself.

Note that the neighborhood vectors are not the absolute position of the elements, but rather a set of relative positions (deltas) with respect to any given site.

There are discrete and continuous variants of Lenia. Let ${\displaystyle \mathbf {x} }$ be a vector in ${\displaystyle \mathbb {R} ^{2}}$ within ${\displaystyle {\mathcal {L}}}$ representing the position of a given site, and ${\displaystyle {\mathcal {N}}}$ be the set of sites neighboring ${\displaystyle \mathbf {x} }$. Both variations comprise two stages:

1. Using a convolution kernel ${\displaystyle \mathbf {K}$ :{\mathcal {N}}\rightarrow S} to compute the potential distribution ${\displaystyle \mathbf {U} ^{t}(\mathbf {x} )=\mathbf {K} *\mathbf {A} ^{t}(\mathbf {x} )}$.
2. Using a growth mapping ${\displaystyle G:[0,1]\rightarrow [-1,1]}$ to compute the final growth distribution ${\displaystyle \mathbf {G} ^{t}(\mathbf {x} )=G(\mathbf {U} ^{t}(\mathbf {x} ))}$.

Once ${\displaystyle \mathbf {G} ^{t}}$ is computed, it is scaled by the chosen time resolution ${\displaystyle \Delta t}$ and added to the original state value:

$${\displaystyle \mathbf {A} ^{t+\Delta t}(\mathbf {x} )={\text{clip}}(\mathbf {A} ^{t}+\Delta t\;\mathbf {G} ^{t}(\mathbf {x} ),\;0,\;1)}$$

Here, the clip function is defined by ${\displaystyle \operatorname {clip} (v,a,b):=\min(\max(u,a),b)}$ .

The local rules are defined as follows for discrete and continuous Lenia:

$${\displaystyle {\begin{aligned}\mathbf {U} ^{t}(\mathbf {x} )&={\begin{cases}\sum _{\mathbf {n} \in {\mathcal {N}}}\mathbf {K(n)} \mathbf {A} ^{t}(\mathbf {x} +\mathbf {n} )\Delta x^{2},&{\text{discrete Lenia}}\\\int _{\mathbf {n} \in {\mathcal {N}}}\mathbf {K(n)} \mathbf {A} ^{t}(\mathbf {x} +\mathbf {n} )dx^{2},&{\text{continuous Lenia}}\end{cases}}\\\mathbf {G} ^{t}(\mathbf {x} )&=G(\mathbf {U} ^{t}(\mathbf {x} ))\\\mathbf {A} ^{t+\Delta t}(\mathbf {x} )&={\text{clip}}(\mathbf {A} ^{t}(\mathbf {x} )+\Delta t\;\mathbf {G} ^{t}(\mathbf {x} ),\;0,\;1)\end{aligned}}}$$

The kernel shell, kernel skeleton, and growth mappings for Lenia.

There are many ways to generate the convolution kernel ${\displaystyle \mathbf {K} }$. The final kernel is the composition of a kernel shell ${\displaystyle K_{C}}$ and a kernel skeleton ${\displaystyle K_{S}}$.

For the kernel shell ${\displaystyle K_{C}}$, Chan gives several functions that are defined radially. Kernel shell functions are unimodal and subject to the constraint ${\displaystyle K_{C}(0)=K_{C}(1)=0}$ (and typically ${\displaystyle K_{C}\left({\frac {1}{2}}\right)=1}$ as well). Example kernel functions include:

$${\displaystyle K_{C}(r)={\begin{cases}\exp \left(\alpha -{\frac {\alpha }{4r(1-r)}}\right),&{\text{exponential}},\alpha =4\\(4r(1-r))^{\alpha },&{\text{polynomial}},\alpha =4\\\mathbf {1} _{\left[{\frac {1}{4}},{\frac {3}{4}}\right]}(r),&{\text{rectangular}}\\\ldots ,&{\text{etc.}}\end{cases}}}$$

Here, ${\displaystyle \mathbf {1} _{A}(r)}$ is the indicator function.

Once the kernel shell has been defined, the kernel skeleton ${\displaystyle K_{S}}$ is used to expand it and compute the actual values of the kernel by transforming the shell into a series of concentric rings. The height of each ring is controlled by a kernel peak vector ${\displaystyle \beta =(\beta _{1},\beta _{2},\ldots ,\beta _{B})\in [0,1]^{B}}$, where ${\displaystyle B}$ is the rank of the parameter vector. Then the kernel skeleton ${\displaystyle K_{S}}$ is defined as

$${\displaystyle K_{S}(r;\beta )=\beta _{\lfloor Br\rfloor }K_{C}(Br{\text{ mod }}1)}$$

The final kernel ${\displaystyle \mathbf {K} (\mathbf {n} )}$ is therefore

$${\displaystyle \mathbf {K} (\mathbf {n} )={\frac {K_{S}(\lVert \mathbf {n} \rVert _{2})}{|K_{S}|}}}$$

such that ${\displaystyle \mathbf {K} }$ is normalized to have an element sum of ${\displaystyle 1}$ and ${\displaystyle \mathbf {K} *\mathbf {A} \in [0,1]}$ (for conservation of mass). ${\displaystyle |K_{S}|=\textstyle \sum _{\mathcal {N}}\displaystyle K_{S}\,\Delta x^{2}}$ in the discrete case, and ${\displaystyle \int _{N}K_{S}\,dx^{2}}$ in the continuous case.

The growth mapping ${\displaystyle G:[0,1]\rightarrow [-1,1]}$, which is analogous to an activation function, may be any function that is unimodal, nonmonotonic, and accepts parameters ${\displaystyle \mu ,\sigma \in \mathbb {R} }$. Examples include

$${\displaystyle G(u;\mu ,\sigma )={\begin{cases}2\exp \left(-{\frac {(u-\mu )^{2}}{2\sigma ^{2}}}\right)-1,&{\text{exponential}}\\2\cdot \mathbf {1} _{[\mu \pm 3\sigma ]}(u)\left(1-{\frac {(u-\mu )^{2}}{9\sigma ^{2}}}\right)^{\alpha }-1,&{\text{polynomial}},\alpha =4\\2\cdot \mathbf {1} _{[\mu \pm \sigma ]}(u)-1,&{\text{rectangular}}\\\ldots ,&{\text{etc.}}\end{cases}}}$$

where ${\displaystyle u}$ is a potential value drawn from ${\displaystyle \mathbf {U} ^{t}}$.

The Game of Life may be regarded as a special case of discrete Lenia with ${\displaystyle R=T=P=1}$. In this case, the kernel would be rectangular, with the function

$${\displaystyle K_{C}(r)=\mathbf {1} _{\left[{\frac {1}{4}},{\frac {3}{4}}\right]}(r)+{\frac {1}{2}}\mathbf {1} _{\left[0,{\frac {1}{4}}\right)}(r)}$$

and the growth rule also rectangular, with ${\displaystyle \mu =0.35,\sigma =0.07}$.