Table 2.

The equations of the DIVA neural network components.

Model Components	Cell Equations	Description of Variables
Speech Sound Map	$P_i\left(t\right)=\{\begin{array}{cc}1\hfill & \mathit{if}\mathit{the}{i}^{\mathit{th}}\mathit{sound}\mathit{is}\mathit{produced}or\mathit{perceived}\hfill \\ 0\hfill & \mathit{otherwise}\hfill \end{array}\phantom{\}}$	Cells are hypothesized to be active during speech production, driving the movement trajectories of the articulators, and during speech perception, when auditory expectations are tuned.
Articulator Velocity Map	${\stackrel{.}{M}}_{\mathit{Feedforward}}\left(t\right)=P\left(t\right)z_{\mathit{PM}}\left(t\right)-M\left(t\right)$	ZPM(t) encodes the feedforward motor commands for the speech sound currently being produced. M(t) specifies the current articulator position. Ṁ Feedforward encodes feedforward articulator velocity commands.
Auditory State Map	$\mathrm{Au}\left(\mathrm{t}\right)=f_{\mathrm{AcAu}}\left(\mathit{Acoust}(t-{\tau }_{\mathit{AcAu}})\right)$	Acoust(t) is the acoustic signal resulting from the current articulator movement. τAcAu is the delay associated with transmitting the auditory signal from the periphery to the cerebral cortex. fAcAu transforms the acoustic signal into a corresponding auditory cortical map representation, Au(t).
Auditory Target Map	$T_{\mathit{Au}}\left(\mathrm{t}\right)=P(t-{\tau }_{\mathit{PAu}})z_{\mathit{PAu}}\left(t\right)$	τPAu is the transmission delay for the signals from the Speech Sound Map to the Auditory Target Map. ZPAu(t) are synaptic weights that encode the auditory expectations, TAu(t), for the current sound being produced.
Auditory Error Map	$\Delta \mathit{Au}\left(t\right)=\mathit{Au}\left(t\right)-T_{\mathit{Au}}\left(t\right)$	ΔAu(t) is the difference between the actual auditory responses and the auditory expectations for the current speech sound.
Somatosensory State Map	$S\left(\mathrm{t}\right)=f_{\mathit{ArS}}\left(\mathit{Artic}(t-{\tau }_{\mathit{ArS}})\right)$	Artic(t) is the state of the articulators resulting from the current articulator movement. τArS is the delay associated with transmitting the articulator state to the cerebral cortex. fArS transforms the articulatory state into the corresponding somatosensory cortical map representation, S(t).
Somatosensory Target Map	$T_s\left(t\right)=P(t-{\tau }_{\mathit{PS}})z_{PS}\left(t\right)$	τPS is the transmission delay for the signals from the Speech Sound Map to the Somatosensory Target Map. zPS(t) are the synaptic weights that encode the somatosensory expectations, TS(t), for the current sound being produced.
Somatosensory Error Map	$\Delta S\left(t\right)=S\left(t\right)-T_S\left(t\right)$	ΔS(t) is the difference between the actual somatosensory responses and the somatosensory expectations for the current speech sound.
Feedback Control Map	${\stackrel{.}{M}}_{\mathit{Feedback}}\left(t\right)=\Delta \mathit{Au}(t-{\tau }_{\mathit{AuM}})z_{\mathit{AuM}}+\Delta S(t-{\tau }_{\mathit{SM}})z_{\mathit{SM}}$	ZAuM and ZSM are synaptic weights that transform directional sensory error signals into corrective motor velocities. τAuM and τSM are cortico-cortical transmission delays. Ṁ Feedback encodes articulator feedback velocity commands.
Initiation Map	$I_i\left(t\right)=\{\begin{array}{cc}1\hfill & \mathit{if}\mathit{the}{i}^{\mathit{th}}\mathit{sound}\mathit{is}\mathit{produced}\hfill \\ 0\hfill & \mathit{otherwise}\hfill \end{array}\phantom{\}}$	Commands from the Articulator Velocity and Position Maps are released to the articulatory synthesizer when the ith cell in the Initiation Map, Ii(t), is non-zero.
Articulator Position Map	$M\left(t\right)=M\left(0\right)+{\alpha }_{\mathit{ff}}{\int }_0^t{\stackrel{.}{M}}_{\mathit{Feedforward}}\left(t\right)I_i\left(t\right)\mathit{dt}+{\alpha }_{\mathit{fb}}{\int }_0^t{\stackrel{.}{M}}_{\mathit{Feedback}}\left(t\right)I_i\left(t\right)\mathit{dt}$	M(0) is the initial configuration of the vocal tract. α fb and α ff are weights relating the contributions of feedback and feedforward control, respectively.