Topologically Defined Neuronal Networks Controlled by Silicon Chips

More documents

Recommendations

Info

CHAPTER 2. NETWORKS OF DEFINED TOPOGRAPHY Figure 2.10: Simple model of two electrically coupled neurons. In this equivalent circuit each cell is represented by a single isopotential compartment. Neuron A is presynaptic, B postsynaptic. overall length is much smaller than λ, that means if they are electrotonically compact; for details refer to the previous subsection. Note that throughout this subsection voltage, capacitance and conductance are somatic quantities; the index ‘s’ used in the previous subsection to distinguish between somatic and neuritic quantities is omitted here. Capital letters denote the respective neuron the variable refers to. Injecting a constant current Iinj into neuron A causes a voltage response VA(t) in the presynaptic cell that is governed by the following differential equation, with Gsyn denoting the conductance of the synapse and V0A the resting potential of neuron A: dVA Iinj = CA dt + GA(VA − V0A) + Gsyn(VA − VB) (2.16) The current flowing across the synapse changes the voltage in the presynaptic neuron B, VB(t), according to the current balance: dVB Gsyn(VA − VB) = CB dt + GB(VB − V0B) (2.17) After transients have decayed, i.e. in the steady state where dVA/dt = dVB/dt = 0, the somatic voltages of A and B are: VA = V0A − GBGsyn(V0A − V0B) GAGB + Gsyn(GA + GB) + GB + Gsyn GAGB + Gsyn(GA + GB) Iinj VB = V0B + GAGsyn(V0A − V0B) GAGB + Gsyn(GA + GB) + Gsyn GAGB + Gsyn(GA + GB) Iinj (2.18) (2.19) Even if Iinj = 0, VA and VB are not equal to the resting potentials if V0A = V0B. This deviation is caused by a small permanent current which flows across the synapse, pulling the membrane voltages towards each other and away from the resting values. All unknown electrical parameters of the simple network in fig. 2.10, except for the membrane capacitances, can be obtained from steady state current measurements. In that case, current balances for each neuron before (index: 0) and after (index: ∞) a current step from Iinj,0 = 0 to Iinj,∞ are given by: 22 0 = GA(VA,0 − V0A) + Gsyn(VA,0 − VB,0) (2.20) Iinj,∞ = GA(VA,∞ − V0A) + Gsyn(VA,∞ − VB,∞) (2.21) Gsyn(VA,0 − VB,0) = GB(VB,0 − V0B) (2.22) Gsyn(VA,∞ − VB,∞) = GB(VB,∞ − V0B) (2.23)
2.4. THEORY PART I: NEURONS, SYNAPSES AND CABLES A is presynaptic and B postsynaptic. Eliminating V0B by subtraction of 2.22 and 2.23 yields an expression which is frequently used in the literature to characterize electrical synapses. The coupling coefficient kA,B is defined as the ratio of the stationary postsynaptic and presynaptic response of the membrane voltage to a hyperpolarizing current step kA,B = △VB △VA = VB,∞ − VB,0 VA,∞ − VA,0 = Gsyn GB + Gsyn (2.24) Although this parameter describes the synapse in its function by quantifying the impact of a presynaptic voltage change on the postsynaptic neuron, it conveys little information about the synapse itself. For a given pair of neurons, kA,B can attain two values, depending on which neuron is postsynaptic with the respective membrane conductance GB. The real parameter characterizing the behavior of an electrical synapse, namely its conductivity Gsyn, is obtained from 2.20 and 2.21 by solving for GA and equating the two terms. Gsyn = VA,0 − V0A VA,∞VB,0 − VA,0VB,∞ − V0A(△VA − △VB) Iinj,∞ (2.25) Dividing 2.22 by 2.23 gives the resting potential of neuron B in terms of the membrane voltages before and after the current step. V0B = VA,∞VB,0 − VA,0VB,∞ △VA − △VB (2.26) Furthermore, from equations 2.20 to 2.23 the following expressions for the membrane conductivities of cell A and B are derived: GA = VB,0 − VA,0 VA,∞VB,0 − VA,0VB,∞ − V0A(△VA − △VB) Iinj,∞ GB = VA,0 − V0A VB,0 − V0B · VA,0 − VB,0 VA,∞VB,0 − VA,0VB,∞ − V0A(△VA − △VB) Iinj,∞ (2.27) (2.28) To calculate Gsyn and the soma conductance, the resting potential of neuron A must be known, too. Obviously, it cannot be obtained from the equation system 2.20 - 2.23 as there are only 4 equations but 5 unknown variables. A second set of current step measurements is necessary to determine V0A, this time with B as the presynaptic and A as the postsynaptic neuron. Exchanging indices A and B in equation 2.26 and inserting the respective voltages yields the desired quantity. Until now, only the steady state case has been considered. The two coupled differential equations 2.16 and 2.17 characterize the dynamic behavior of a synaptically connected cell pair. From the system’s characteristic polynomial two expressions can be deduced, describing the time course of the membrane voltage in both neurons, each being the sum of two mono-exponential transients and a constant; for the solutions refer to [79]. Detailed model The isopotential model above neglects voltage drops in the neurites and therefore gives only approximate values of Gsyn. To obtain the exact synaptic conductivity, the cable properties of the neurites have to be taken into account, resulting in the equivalent circuit shown in fig. 2.11. As before, the synapse is represented by a conductance, but this time it is located at the end of neurites, whose electrical behavior is described with the somatic shunt cable model, discussed in subsection 2.4.1. In analogy to the isopotential model Gsyn is defined by the current flowing across the synapse Isyn and the membrane 23
Page 1: Max-Planck-Institut für Biochemie
Page 5 and 6: Contents 1 Introduction 1 1.1 Why b
Page 7 and 8: Chapter 1 Introduction This first c
Page 9 and 10: 1.3. OUTLINE OF THIS THESIS displac
Page 11 and 12: Chapter 2 Networks of defined topog
Page 13 and 14: 2.1. HOW TO CONTROL NEURONAL OUTGRO
Page 15 and 16: 2.1. HOW TO CONTROL NEURONAL OUTGRO
Page 17 and 18: 2.2. CELL CULTURE Figure 2.4: A: Po
Page 19 and 20: 2.2. CELL CULTURE Figure 2.5: A: Do
Page 21 and 22: 2.3. BUILDING DEFINED NETWORKS - TE
Page 23 and 24: 2.3. BUILDING DEFINED NETWORKS - TE
Page 25 and 26: 2.4. THEORY PART I: NEURONS, SYNAPS
Page 27: 2.4. THEORY PART I: NEURONS, SYNAPS
Page 31 and 32: 2.4. THEORY PART I: NEURONS, SYNAPS
Page 33 and 34: y voltage-gated channels, as outlin
Page 35 and 36: 2.5. RESULTS Figure 2.14: A: Neuron
Page 37 and 38: 2.5. RESULTS DM (without diluting i
Page 39 and 40: 2.5. RESULTS Based on the assumptio
Page 41 and 42: 2.5. RESULTS Figure 2.21: Electrica
Page 43 and 44: 2.5. RESULTS Figure 2.22: A: Histog
Page 45 and 46: 2.5. RESULTS Figure 2.24: Three neu
Page 47 and 48: 2.5. RESULTS Figure 2.26: Statistic
Page 49 and 50: Chapter 3 Single neurons on chips A
Page 51 and 52: 3.2 Theory part II: Field-effect tr
Page 53 and 54: 3.2. THEORY PART II: FIELD-EFFECT T
Page 55 and 56: A C 3.2. THEORY PART II: FIELD-EFFE
Page 57 and 58: 3.3. THE TRANSISTOR-STIMULATOR CHIP
Page 61 and 62: Silicon dioxide 3.3. THE TRANSISTOR
Page 63 and 64: A C B D 3.3. THE TRANSISTOR-STIMULA
Page 69 and 70: 3.4. MEASUREMENT SETUP Figure 3.11:
Page 71 and 72: 3.4. MEASUREMENT SETUP onboard memo
Page 73 and 74: 3.5. THEORY PART III: NEURON-SILICO
Page 75 and 76: characterizing the dynamics of the
Page 77 and 78: 3.5. THEORY PART III: NEURON-SILICO
Page 79 and 80:
3.6. NEURON-SILICON INTERFACE: EXPE
Page 81 and 82:
Page 83 and 84:
demonstrated by the examples in fig
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Chapter 4 Defined networks on chips
Page 91 and 92:
4.2. RESULTS Figure 4.2: A: Microgr
Page 93 and 94:
4.2. RESULTS 2 postsynaptic action
Page 95 and 96:
4.2. RESULTS Since neuron 1 could a
Page 97 and 98:
4.2. RESULTS Figure 4.5: A: Transis
Page 99 and 100:
4.3. FINAL DISCUSSION Data in diagr
Page 101 and 102:
4.3. FINAL DISCUSSION Some very har
Page 103 and 104:
Chapter 5 Conclusion and outlook Th
Page 105 and 106:
Appendix A Cell culture protocols a
Page 107 and 108:
ethanol and 30% water during UV-ste
Page 109 and 110:
For two-layer structures, steps 3 t
Page 111 and 112:
The flatband voltage VF B and elect
Page 113 and 114:
Appendix D Abbreviations ABS : anti
Page 115 and 116:
Bibliography [1] R. W. Aldrich, P.
Page 117 and 118:
BIBLIOGRAPHY [32] P. Fromherz. Neur
Page 119 and 120:
BIBLIOGRAPHY [69] C. D. McCaig, A.
Page 121 and 122:
BIBLIOGRAPHY [105] N. I. Syed, A. G
Page 123:
Ich danke PROF. DR. P. FROMHERZ fü
show all

Topologically Defined Neuronal Networks Controlled by Silicon Chips

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?