# Theoretical Aspects of Resting-State Cardiomyocyte Communication for Multi-Nodal Nano-Actuator Pacemakers

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## Abstract

**:**

## 1. Introduction

## 2. Subthreshold Cardiac Communication System

#### 2.1. Transfer Impedance

#### 2.2. Noiseless Input-Output Relation

## 3. Noise in the Subthreshold Cardiac Communication System

#### 3.1. Encoding Noise

#### 3.2. Membrane-Related Noise

#### 3.2.1. Voltage-Gated Channel Noise

#### 3.2.2. Shot Noise

#### 3.2.3. Thermal Noise

#### 3.3. Noisy Input-Output Relation

## 4. Numerical Simulations

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Membrane Linearization

#### Appendix A.2. Derivation of Current Noise PSD

## References

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**Figure 1.**(

**a**) Nodes from different parts of the heart produce diverse action potential signals. The composition of action potentials generates an ECG signal. (

**b**) Nano-actuator pacemaker network in the heart ventricle: multiple nano-actuators are distributed in the ventricle and are coordinated by the gateway/hub. The nano-actuators are envisioned to share information to enhance their abilities [1]. The figure is adapted from an existing image provided by Servier Medical Art by Servier, licensed under a Creative Commons Attribution 3.0 Unported License.

**Figure 2.**Encoding subthreshold membrane potentials are envisioned to be transmitted over cardiac cellular infrastructure within time bins between consecutive action potentials.

**Figure 3.**Communication channel in the subthreshold cardiac communication system. (

**a**) The general representation of the subthreshold cardiac communication channel as a one-dimensional cable. (

**b**) The linearized membrane circuit corresponding to the shaded block in Figure 3a. (1)-segment consists of the capacitor derived from the bilayer membrane; (2)-segment consists of passive components derived from the voltage-gated sodium channels; (3)-segment consists of passive components derived from the voltage-gated potassium channels; (4)-segment consists of passive components derived from the voltage-gated slow inward current which mainly contains calcium channels; (5)-segment consists of the resistor derived from the plateau potassium current and background current.

**Figure 7.**The eye diagram of subthreshold cardiac communication system. The stimulation amplitude is 3 nA, and the transmission rate is 5 bit/s. Blue curves correspond to noiseless scenarios; orange curves correspond to noisy scenarios. (

**a**,

**b**): The transmission distance corresponds to the one-cell length. (

**c**,

**d**): The transmission distance corresponds to the four-cell length. (

**e**,

**f**): The transmission distance corresponds to the eight-cell length. (

**g**,

**h**): The transmission distance corresponds to the twelve-cell length.

**Figure 8.**Transmission of bit sequence over ten cardiomyocytes with stimulation signal amplitude of 3 nA. The transmission bin of 2 seconds has been selected to demonstrate the system performance. (

**a**) Sample bit sequence and associated symbols. (

**b**) Transmitted signal. (

**c**) Received signal. (

**d**) Decoded signal.

Parameter | Description | Unit |
---|---|---|

${S}_{Tx}\left(jf\right)$ | Input current PSD | $\mathsf{\mu}$A${}^{2}$/Hz |

${z}_{m}\left(jf\right)$ | Membrane impedance (per unit length) | k$\mathsf{\Omega}\xb7$ cm |

${Z}_{m}\left(jf\right)$ | Resistivity of membrane | k$\mathsf{\Omega}\xb7$ cm${}^{2}$ |

${z}_{l}$ | Equivalent longitudinal impedance (per unit length) | k$\mathsf{\Omega}$/cm |

${Z}_{l}$ | Equivalent longitudinal resistivity | k$\mathsf{\Omega}\xb7$ cm |

$Z(x,jf)$ | Transfer impedance | k$\mathsf{\Omega}$ |

${S}_{Rx}\left(jf\right)$ | Output voltage PSD | mV${}^{2}$/Hz |

${i}_{Tx}\left(t\right)$ | Input current | $\mathsf{\mu}$A |

${\tilde{i}}_{Tx}\left(t\right)$ | Input current corrupted with input-dependent noise | $\mathsf{\mu}$A |

${i}_{{N}_{1}}\left(t\right)$ | Input-dependent noise current | $\mathsf{\mu}$A |

${S}_{{N}_{1}}\left(jf\right)$ | Current PSD of input-dependent noise current | $\mathsf{\mu}$A${}^{2}$/Hz |

${\tilde{S}}_{Tx}\left(jf\right)$ | Current PSD of the input corrupted with input-dependent noise | $\mathsf{\mu}$A${}^{2}$/Hz |

${\tilde{S}}_{{N}_{2}}^{1}\left(jf\right)$ | Current PSD of voltage-gated channel noise | $\mathsf{\mu}$A${}^{2}$/Hz/cm |

${\tilde{S}}_{{N}_{2}}^{2}\left(jf\right)$ | Current PSD of shot noise | $\mathsf{\mu}$A${}^{2}$/Hz/cm |

${\tilde{S}}_{{N}_{2}}^{3}\left(jf\right)$ | Current PSD of thermal noise | $\mathsf{\mu}$A${}^{2}$/Hz/cm |

${\tilde{S}}_{\mathrm{K}}\left(jf\right)$ | Current PSD of potassium ions | $\mathsf{\mu}$A${}^{2}$/Hz/cm |

${\tilde{S}}_{\mathrm{Na}}\left(jf\right)$ | Current PSD of sodium ions | $\mathsf{\mu}$A${}^{2}$/Hz/cm |

${\tilde{S}}_{\mathrm{Ca}}\left(jf\right)$ | Current PSD of calcium ions | $\mathsf{\mu}$A${}^{2}$/Hz/cm |

${\tilde{S}}_{N{2}_{u}}\left(jf\right)$ | Current PSD of membrane-related noise | $\mathsf{\mu}$A${}^{2}$/Hz/cm |

${\tilde{S}}_{{N}_{2}}(x,jf)$ | Voltage PSD of membrane-related noise | mV${}^{2}$/Hz |

${\tilde{S}}_{Rx}(x,jf)$ | Output noisy voltage PSD | mV${}^{2}$/Hz |

Parameter | Description | Value | Unit |
---|---|---|---|

${C}_{m}$ | Specific membrane capacitance | 1 | $\mathsf{\mu}$F/cm${}^{2}$ |

${E}_{\mathrm{Na}}$ | Reversal potential | 54.4 | mV |

${E}_{\mathrm{K}}$ | Reversal potential | −77 | mV |

${E}_{si}$ | Reversal potential | 40 | mV |

${\gamma}_{{\mathrm{Na}}^{+}}$ | Channel conductance | 15 | pS |

${\eta}_{\mathrm{Na}}^{patch}$ | Channel density | 1–16 | /$\mathsf{\mu}$m${}^{2}$ |

${\gamma}_{\mathrm{Ca}}$ | Channel conductance | 9–25 | pS |

${\eta}_{\mathrm{Ca}}^{patch}$ | Channel density | 0.5–5 | /$\mathsf{\mu}$m${}^{2}$ |

${\gamma}_{{\mathrm{K}}^{+}}$ | Channel conductance | 1.6 | pS |

${\eta}_{\mathrm{K}}^{patch}$ | Channel density | 0.7 | /$\mathsf{\mu}$m${}^{2}$ |

${Z}_{l}$ | Equivalent longitudinal resistivity | 600 | $\mathsf{\Omega}\xb7$cm |

${S}_{V}$ | Surface-to-volume ratio | 4440 | cm${}^{-1}$ |

${L}_{cell}$ | Cell length | 100 | $\mathsf{\mu}$m |

a | Cell radius | 10 | $\mathsf{\mu}$m |

Parameter | Value | Unit |
---|---|---|

${R}_{\mathrm{Na}}$ | 1.13 × 10${}^{7}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${r}_{m}$ | −1.64 × 10${}^{5}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${L}_{m}$ | −0.98 | k$\mathsf{\Omega}\xb7$s·cm${}^{2}$ |

${r}_{h}$ | 1.92 × 10${}^{7}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${L}_{h}$ | 7.74 × 10${}^{4}$ | k$\mathsf{\Omega}\xb7$s·cm${}^{2}$ |

${r}_{j}$ | 3.13 × 10${}^{7}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${L}_{j}$ | 5.32 × 10${}^{5}$ | k$\mathsf{\Omega}\xb7$s·cm${}^{2}$ |

${R}_{\mathrm{K}}$ | 2.13 × 10${}^{3}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${r}_{X}$ | −3.11 × 10${}^{3}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${L}_{X}$ | −711.64 | k$\mathsf{\Omega}\xb7$s·cm${}^{2}$ |

${R}_{{X}_{i}}$ | 2.32 × 10${}^{6}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${R}_{\mathrm{Ca}}$ | 160.35 | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${r}_{d}$ | −15.29 | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${L}_{d}$ | −0.13 | k$\mathsf{\Omega}\xb7$s·cm${}^{2}$ |

${r}_{f}$ | 3.31 × 10${}^{5}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

${L}_{f}$ | 1.76 × 10${}^{4}$ | k$\mathsf{\Omega}\xb7$s·cm${}^{2}$ |

${R}_{o}$ | 6.11 × 10${}^{8}$ | k$\mathsf{\Omega}\xb7$cm${}^{2}$ |

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**MDPI and ACS Style**

Lu, P.; Veletić, M.; Bergsland, J.; Balasingham, I.
Theoretical Aspects of Resting-State Cardiomyocyte Communication for Multi-Nodal Nano-Actuator Pacemakers. *Sensors* **2020**, *20*, 2792.
https://doi.org/10.3390/s20102792

**AMA Style**

Lu P, Veletić M, Bergsland J, Balasingham I.
Theoretical Aspects of Resting-State Cardiomyocyte Communication for Multi-Nodal Nano-Actuator Pacemakers. *Sensors*. 2020; 20(10):2792.
https://doi.org/10.3390/s20102792

**Chicago/Turabian Style**

Lu, Pengfei, Mladen Veletić, Jacob Bergsland, and Ilangko Balasingham.
2020. "Theoretical Aspects of Resting-State Cardiomyocyte Communication for Multi-Nodal Nano-Actuator Pacemakers" *Sensors* 20, no. 10: 2792.
https://doi.org/10.3390/s20102792