4._CRD Discussion_NN1 :_PUN From_PRP Electrodynamics_NN2 to_PRP Biophysics_NP0 and_CJC Back_AVP 
A_AT0 fundamental_AJ0 question_NN1 arises_VVZ with_PRP the_AT0 updated_AJ0 version_NN1 of_PRF Maxwell_NP0 's_POS equations_NN2 ._SENT How_AVQ is_VBZ the_AT0 phenomenon_NN1 of_PRF polarization_NN1 included_VVN in_PRP Equation_NN1 (_PUL 11_CRD )_PUR and_CJC Equation_NN1 (_PUL 14_CRD )_PUR ?_SENT 
To_TO0 answer_VVI this_DT0 question_NN1 ,_PUN we_PNP first_ORD need_VVB a_AT0 general_AJ0 paradigm_NN1 to_TO0 define_VVI polarization_NN1 ,_PUN even_AV0 when_CJS dielectrics_NN2 are_VBB far_AV0 from_PRP ideal_NN1 ,_PUN when_CJS they_PNP might_VM0 be_VBI time_NN1 and_CJC frequency_NN1 dependent_AJ0 ,_PUN and_CJC voltage_NN1 dependent_AJ0 as_CJS well_AV0 ._SENT We_PNP need_VVB a_AT0 paradigm_NN1 that_CJT describes_VVZ how_AVQ the_AT0 charge_NN1 distribution_NN1 varies_VVZ with_PRP the_AT0 electric_AJ0 field_NN1 in_PRP as_AV0 general_AJ0 a_AT0 system_NN1 as_CJS possible_AJ0 ,_PUN including_PRP systems_NN2 with_PRP charge_NN1 movement_NN1 driven_VVN by_PRP forces_NN2 not_XX0 in_PRP the_AT0 Maxwell_NP0 equations_NN2 at_PRP all_DT0 ,_PUN such_DT0 as_PRP convection_NN1 and_CJC diffusion_NN1 ._SENT 
It_PNP seems_VVZ obvious_AJ0 that_CJT a_AT0 general_AJ0 paradigm_NN1 cannot_NN1 be_VBI found_VVN ._SENT After_PRP all_DT0 the_AT0 motions_NN2 of_PRF matter_NN1 in_PRP response_NN1 to_PRP a_AT0 change_NN1 in_PRP electric_AJ0 field_NN1 are_VBB more_AV0 or_CJC less_AV0 as_CJS complex_AJ0 as_CJS the_AT0 motions_NN2 of_PRF matter_NN1 itself_PNX !_SENT Nonetheless_AV0 ,_PUN a_AT0 paradigm_NN1 of_PRF that_DT0 may_VM0 be_VBI helpful_AJ0 in_PRP many_DT0 cases_NN2 has_VHZ been_VBN in_PRP use_NN1 for_PRP many_DT0 years_NN2 ,_PUN even_AV0 if_CJS it_PNP is_VBZ not_XX0 perfectly_AV0 general_AJ0 ._SENT 
This_DT0 problem_NN1 has_VHZ been_VBN addressed_VVN in_PRP membrane_NN1 biophysics_NN2 ._SENT A_AT0 community_NN1 of_PRF scholars_NN2 has_VHZ studied_VVN the_AT0 nonlinear_AJ0 currents_NN2 that_CJT control_VVB the_AT0 opening_NN1 of_PRF voltage_NN1 sensitive_AJ0 protein_NN1 channels_NN2 for_PRP nearly_AV0 fifty_CRD years_NN2 ,_PUN inspired_VVN by_PRP [_PUL 6_CRD ]_PUR ._SENT They_PNP have_VHB developed_VVN protocols_NN2 that_CJT may_VM0 be_VBI useful_AJ0 in_PRP other_AJ0 systems_NN2 ,_PUN as_CJS they_PNP have_VHB been_VBN in_PRP biophysics_NN2 ._SENT Schneider_NP0 and_CJC Chandler_NP0 followed_VVD by_PRP Bezanilla_NP0 and_CJC Armstrong_NP0 are_VBB responsible_AJ0 for_PRP this_DT0 paradigm_NN1 ,_PUN more_AV0 than_CJS anyone_PNI else_AV0 ._SENT 
The_AT0 basic_AJ0 setup_NN1 used_VVN in_PRP these_DT0 experiments_NN2 is_VBZ that_DT0 of_PRF an_AT0 electrochemical_AJ0 cell_NN1 modified_VVN to_TO0 deal_VVI with_PRP a_AT0 cylindrical_AJ0 cell_NN1 as_CJS shown_VVN in_PRP Figure_NN1 1._CRD Membrane_NN1 potential_NN1 is_VBZ measured_VVN across_PRP a_AT0 biological_AJ0 membrane_NN1 ,_PUN with_PRP defined_AJ0 concentrations_NN2 on_PRP both_DT0 sides_NN2 of_PRF the_AT0 membrane_NN1 ._SENT Current_NN1 is_VBZ applied_VVN through_PRP electrodes_NN2 to_TO0 control_VVI the_AT0 potential_NN1 ,_PUN in_PRP the_AT0 classical_AJ0 voltage_NN1 clamp_NN1 set_VVD up_AVP of_PRF Cole_NP0 and_CJC Hodgkin_NP0 ,_PUN Huxley_NP0 ,_PUN and_CJC Katz_NP0 ._SENT It_PNP is_VBZ best_AJS to_TO0 apply_VVI that_DT0 current_AJ0 in_PRP electrodes_NN2 different_AJ0 from_PRP those_DT0 that_DT0 record_NN1 membrane_NN1 potential_NN1 using_VVG a_AT0 so-called_AJ0 four_CRD electrode_NN1 setup_NN1 ,_PUN like_PRP those_DT0 described_VVN in_PRP textbooks_NN2 of_PRF electrochemistry_NN1 ._SENT 
I_PNP propose_VVB using_VVG the_AT0 operational_AJ0 definition_NN1 of_PRF '_PUQ gating_AJ0 current_NN1 '_POS used_AJ0 to_TO0 define_VVI nonlinear_AJ0 ,_PUN time_NN1 and_CJC voltage_NN1 dependent_AJ0 polarization_NN1 by_PRP biophysicists_NN2 as_PRP a_AT0 useful_AJ0 setup_NN1 and_CJC definition_NN1 of_PRF many_DT0 types_NN2 of_PRF polarization_NN1 ._SENT Obviously_AV0 ,_PUN this_DT0 definition_NN1 is_VBZ not_XX0 general_AJ0 ,_PUN but_CJC the_AT0 hope_NN1 is_VBZ that_CJT it_PNP may_VM0 be_VBI generally_AV0 useful_AJ0 ._SENT 
The_AT0 basic_AJ0 idea_NN1 is_VBZ to_TO0 apply_VVI a_AT0 set_NN1 of_PRF step_NN1 functions_NN2 of_PRF potential_NN1 across_PRP the_AT0 system_NN1 —_PUN in_PRP biology_NN1 across_PRP the_AT0 membrane_NN1 —_PUN and_CJC observe_VVB the_AT0 currents_NN2 that_CJT flow_VVB ._SENT The_AT0 currents_NN2 observed_VVN are_VBB transients_NN2 that_CJT decline_VVB to_PRP a_AT0 steady_AJ0 value_NN1 ,_PUN often_AV0 to_PRP near_AJ0 zero_NN1 after_PRP a_AT0 reasonable_AJ0 (_PUL biologically_AV0 relevant_AJ0 )_PUR time_NN1 ._SENT The_AT0 measured_AJ0 currents_NN2 are_VBB perfectly_AV0 reproducible_AJ0 ._SENT If_CJS a_AT0 pulse_NN1 is_VBZ applied_VVN ,_PUN the_AT0 charge_NN1 moved_VVN (_PUL the_AT0 integral_AJ0 of_PRF the_AT0 current_NN1 )_PUR can_VM0 be_VBI measured_VVN when_CJS the_AT0 voltage_NN1 step_NN1 is_VBZ applied_VVN ._SENT The_AT0 integration_NN1 goes_VVZ on_AVP until_PRP t1_NN1 when_CJS the_AT0 current_AJ0 ileak_NN1 is_VBZ nearly_AV0 independent_AJ0 of_PRF time_NN1 ,_PUN often_AV0 nearly_AV0 zero_CRD ._SENT That_DT0 integral_AJ0 is_VBZ called_VVN the_AT0 ON_PRP charge_NN1 QON_NN1 ._SENT 
When_CJS the_AT0 voltage_NN1 is_VBZ returned_VVN to_PRP its_DPS initial_AJ0 value_NN1 (_PUL the_AT0 value_NN1 that_CJT was_VBD present_AJ0 before_PRP the_AT0 ON_PRP pulse_NN1 )_PUR ,_PUN another_DT0 current_NN1 is_VBZ observed_VVN that_CJT often_AV0 has_VHZ quite_AV0 different_AJ0 time_NN1 course_NN1 ,_PUN much_AV0 more_AV0 so_AV0 than_CJS in_PRP Figure_NN1 1._CRD The_AT0 integral_AJ0 of_PRF that_DT0 current_NN1 is_VBZ the_AT0 OFF_AVP charge_NN1 QOFF_NN1 ._SENT 
If_CJS QON_NP0 =_NP0 QOFF_NP0 ,_PUN and_CJC the_AT0 physical_AJ0 processes_NN2 involved_VVN depend_VVB fundamentally_AV0 on_PRP potential_AJ0 and_CJC not_XX0 its_DPS time_NN1 derivative_NN1 ,_PUN the_AT0 biophysical_AJ0 paradigm_NN1 is_VBZ likely_AJ0 to_TO0 be_VBI useful_AJ0 ._SENT In_PRP other_AJ0 cases_NN2 ,_PUN another_DT0 paradigm_NN1 is_VBZ needed_VVN ._SENT If_CJS the_AT0 current_NN1 produced_VVN by_PRP the_AT0 step_NN1 in_PRP potential_NN1 is_VBZ in_PRP fact_NN1 actually_AV0 transient_AJ0 ,_PUN the_AT0 steady_AJ0 current_NN1 will_VM0 be_VBI what_DTQ it_PNP was_VBD before_PRP the_AT0 voltage_NN1 step_NN1 was_VBD applied_VVN ._SENT The_AT0 transient_AJ0 will_VM0 disappear_VVI with_PRP time_NN1 as_PRP the_AT0 word_NN1 '_POS transient_AJ0 '_PUQ implies_VVZ ._SENT In_PRP that_DT0 case_NN1 it_PNP seems_VVZ that_CJT the_AT0 biophysical_AJ0 paradigm_NN1 is_VBZ not_XX0 only_AV0 useful_AJ0 but_CJC may_VM0 even_AV0 provide_VVI a_AT0 unique_AJ0 definition_NN1 of_PRF gating_VVG current_AJ0 and_CJC the_AT0 corresponding_AJ0 polarization_NN1 ._SENT 
Gating_NN1 current_AJ0 as_CJS measured_VVN in_PRP biophysical_AJ0 experiments_NN2 depends_VVZ on_PRP the_AT0 membrane_NN1 voltage_NN1 before_PRP the_AT0 step_NN1 ,_PUN as_CJS well_AV0 as_CJS the_AT0 voltage_NN1 just_AV0 after_PRP and_CJC during_PRP the_AT0 step_NN1 ._SENT It_PNP also_AV0 depends_VVZ separately_AV0 on_PRP the_AT0 voltage_NN1 after_PRP the_AT0 step_NN1 ,_PUN although_CJS Figure_NN1 1_CRD does_VDZ not_XX0 illustrate_VVI the_AT0 dependence_NN1 documented_VVN in_PRP the_AT0 literature_NN1 ._SENT The_AT0 voltage_NN1 and_CJC time_NN1 dependence_NN1 arises_VVZ from_PRP the_AT0 molecular_AJ0 motions_NN2 underlying_VVG the_AT0 gating_AJ0 current_NN1 ._SENT The_AT0 voltage_NN1 and_CJC time_NN1 dependence_NN1 defines_VVZ the_AT0 mean_AJ0 molecular_AJ0 motions_NN2 and_CJC is_VBZ called_VVN '_PUQ the_AT0 gating_AJ0 current_NN1 '_PUQ in_PRP the_AT0 biophysics_NN2 literature_NN1 ._SENT 
If_CJS the_AT0 ON_PRP charge_NN1 is_VBZ found_VVN experimentally_AV0 to_TO0 equal_VVI the_AT0 OFF_AVP charge_NN1 ,_PUN for_PRP a_AT0 variety_NN1 of_PRF pulse_NN1 sizes_NN2 and_CJC range_NN1 of_PRF experimental_AJ0 conditions_NN2 ,_PUN the_AT0 current_NN1 is_VBZ said_VVN to_TO0 arise_VVI in_PRP a_AT0 nonlinear_AJ0 (_PUL i.e._AV0 ,_PUN voltage_NN1 dependent_AJ0 )_PUR polarization_NN1 capacitance_NN1 and_CJC is_VBZ interpreted_VVN as_PRP the_AT0 movement_NN1 of_PRF charged_AJ0 groups_NN2 in_PRP the_AT0 electric_AJ0 field_NN1 ._SENT The_AT0 charged_AJ0 groups_NN2 move_VVB to_PRP one_CRD location_NN1 after_PRP the_AT0 ON_PRP pulse_NN1 ,_PUN and_CJC return_VVB to_PRP their_DPS original_AJ0 location_NN1 following_VVG the_AT0 OFF_AVP pulse_NN1 ._SENT The_AT0 charge_NN1 is_VBZ called_VVN '_PUQ gating_AJ0 charge_NN1 '_PUQ ,_PUN and_CJC the_AT0 current_NN1 that_CJT carries_VVZ the_AT0 charge_NN1 is_VBZ called_VVN '_PUQ gating_AJ0 current_NN1 '_POS ._SENT 
The_AT0 macroscale_AJ0 current_NN1 observed_VVN in_PRP the_AT0 set-up_NN1 is_VBZ equal_AJ0 to_PRP the_AT0 sum_NN1 of_PRF the_AT0 micro_NN1 (_PUL actually_AV0 atomic_AJ0 scale_NN1 )_PUR currents_NN2 carried_VVN by_PRP the_AT0 charged_AJ0 groups_NN2 inside_PRP a_AT0 channel_NN1 protein_NN1 ,_PUN even_AV0 though_CJS the_AT0 recording_NN1 electrodes_NN2 are_VBB remote_AJ0 from_PRP the_AT0 protein_NN1 ._SENT Indeed_AV0 ,_PUN there_EX0 might_VM0 be_VBI 1018_CRD charged_AJ0 atoms_NN2 (_PUL ions_NN2 )_PUR between_PRP the_AT0 electrodes_NN2 and_CJC the_AT0 protein_NN1 ._SENT 
The_AT0 currents_NN2 in_PRP the_AT0 electrodes_NN2 and_CJC the_AT0 channel_NN1 protein_NN1 are_VBB equal_AJ0 because_CJS the_AT0 setup_NN1 is_VBZ designed_VVN to_TO0 be_VBI an_AT0 unbranched_AJ0 one-dimensional_AJ0 circuit_NN1 with_PRP everything_PNI in_PRP series_NN0 ._SENT In_PRP a_AT0 one-dimensional_AJ0 series_NN0 setup_VVB the_AT0 total_AJ0 current_NN1 is_VBZ equal_AJ0 everywhere_AV0 in_PRP the_AT0 series_NN0 system_NN1 at_PRP any_DT0 one_CRD time_NN1 ,_PUN even_AV0 though_CJS the_AT0 total_AJ0 current_NN1 varies_VVZ significantly_AV0 with_PRP time_NN1 ._SENT The_AT0 Maxwell_NP0 equations_NN2 guarantee_VVB spatial_AJ0 uniformity_NN1 of_PRF total_AJ0 current_NN1 (_PUL including_PRP the_AT0 ethereal_AJ0 current_AJ0 ε0∂E_NN1 /_PUN ∂t_NN1 )_PUR independent_AJ0 of_PRF the_AT0 microphysics_NN1 of_PRF movement_NN1 of_PRF charge_NN1 (_PUL with_PRP mass_NN1 )_PUR :_PUN Figure_NN1 2_CRD of_PRF [_PUL 73_CRD ]_PUR ,_PUN and_CJC [_PUL 43_CRD ,_PUN 75_CRD ,_PUN 87_CRD ]_PUR ._SENT The_AT0 equality_NN1 of_PRF current_NN1 can_VM0 be_VBI checked_VVN by_PRP measuring_VVG current_AJ0 in_PRP different_AJ0 locations_NN2 in_PRP the_AT0 experiment_NN1 ._SENT The_AT0 spatial_AJ0 equality_NN1 of_PRF current_AJ0 needs_NN2 also_AV0 to_TO0 be_VBI checked_VVN in_PRP simulations_NN2 as_CJS in_PRP [_PUL 18_CRD ,_PUN 21_CRD ,_PUN 88_CRD ]_PUR because_CJS tiny_AJ0 inadvertent_AJ0 errors_NN2 in_PRP numerical_AJ0 procedures_NN2 or_CJC coding_NN1 can_VM0 produce_VVI substantial_AJ0 deviations_NN2 from_PRP spatial_AJ0 equality_NN1 and_CJC thus_AV0 misleading_AJ0 artifacts_NN2 ._SENT Imposing_AJ0 periodic_AJ0 boundary_NN1 conditions_NN2 on_PRP nonperiodic_AJ0 systems_NN2 is_VBZ another_DT0 possible_AJ0 source_NN1 of_PRF such_DT0 artifacts_NN2 ._SENT 
If_CJS the_AT0 currents_NN2 reach_VVB a_AT0 steady_AJ0 value_NN1 independent_AJ0 of_PRF time_NN1 ,_PUN but_CJC not_XX0 equal_AJ0 to_PRP zero_NN1 ,_PUN as_CJS in_PRP Figure_NN1 1_CRD ,_PUN the_AT0 signal_NN1 is_VBZ not_XX0 transient_AJ0 ,_PUN in_PRP the_AT0 strict_AJ0 meaning_NN1 of_PRF the_AT0 word_NN1 ._SENT In_PRP biophysics_NN2 ,_PUN the_AT0 steady_AJ0 current_AJ0 ileak_NN1 is_VBZ then_AV0 usually_AV0 considered_VVN to_TO0 flow_VVI in_PRP a_AT0 resistive_AJ0 path_NN1 that_DT0 is_VBZ time_NN1 independent_AJ0 ,_PUN but_CJC perhaps_AV0 voltage_NN1 dependent_AJ0 ,_PUN in_PRP parallel_NN1 with_PRP the_AT0 path_NN1 or_CJC device_NN1 in_PRP which_DTQ the_AT0 gating_NN1 charges_NN2 QON_NP0 and_CJC QOFF_NP0 flow_NN1 ._SENT If_CJS the_AT0 current_NN1 does_VDZ not_XX0 reach_VVI a_AT0 steady_AJ0 value_NN1 ,_PUN or_CJC if_CJS the_AT0 areas_NN2 in_PRP Figure_NN1 2_CRD are_VBB not_XX0 equal_AJ0 ,_PUN the_AT0 currents_NN2 are_VBB not_XX0 considered_VVN '_PUQ capacitive_AJ0 '_PUQ and_CJC are_VBB interpreted_VVN as_PRP those_DT0 through_PRP a_AT0 time_NN1 and_CJC voltage_NN1 dependent_AJ0 '_POS resistor_NN1 '_PUQ ._SENT This_DT0 is_VBZ a_AT0 biological_AJ0 and_CJC biophysical_AJ0 assumption_NN1 ._SENT It_PNP is_VBZ not_XX0 a_AT0 physical_AJ0 or_CJC mathematical_AJ0 necessity_NN1 ._SENT Thus_AV0 ,_PUN it_PNP is_VBZ important_AJ0 to_TO0 investigate_VVI the_AT0 properties_NN2 of_PRF the_AT0 currents_NN2 through_PRP the_AT0 resistive_AJ0 path_NN1 —_PUN e.g._AV0 ,_PUN those_DT0 that_CJT are_VBB not_XX0 transient_AJ0 and_CJC do_VDB not_XX0 return_VVI to_PRP zero_CRD and_CJC those_DT0 that_CJT make_VVB QON≠QOFF_NN1 by_PRP independent_AJ0 methods_NN2 to_TO0 see_VVI if_CJS they_PNP are_VBB time_NN1 independent_AJ0 ._SENT In_PRP biophysics_NN2 ,_PUN currents_NN2 can_VM0 be_VBI done_VDN by_PRP blocking_VVG the_AT0 resistive_AJ0 path_NN1 with_PRP drugs_NN2 ,_PUN or_CJC with_PRP mutations_NN2 of_PRF the_AT0 channel_NN1 protein_NN1 ._SENT If_CJS the_AT0 resistive_AJ0 currents_NN2 are_VBB not_XX0 time_NN1 independent_AJ0 ,_PUN the_AT0 definition_NN1 of_PRF QON_NN1 and_CJC QOFF_NN1 in_PRP Figure_NN1 1_CRD needs_VVZ to_TO0 be_VBI changed_VVN ._SENT Indeed_AV0 ,_PUN experiments_NN2 of_PRF another_DT0 type_NN1 must_VM0 be_VBI designed_VVN that_CJT allow_VVB separation_NN1 of_PRF polarization_NN1 from_PRP conduction_NN1 currents_NN2 ._SENT The_AT0 simplest_AJS version_NN1 of_PRF the_AT0 biophysics_NN2 paradigm_NN1 then_AV0 needs_VVZ to_TO0 be_VBI extended_VVN ._SENT 
Clearly_AV0 ,_PUN this_DT0 approach_NN1 will_VM0 only_AV0 work_VVI if_CJS step_NN1 functions_NN2 can_VM0 reveal_VVI all_DT0 the_AT0 properties_NN2 of_PRF the_AT0 underlying_AJ0 mechanism_NN1 ._SENT If_CJS the_AT0 underlying_AJ0 mechanisms_NN2 depend_VVB on_PRP the_AT0 time_NN1 rate_NN1 of_PRF change_NN1 of_PRF voltage_NN1 ,_PUN step_NN1 functions_NN2 are_VBB clearly_AV0 insufficient_AJ0 because_CJS ∂V_NP0 /_PUN ∂t_NP0 ,_PUN is_VBZ zero_NN1 or_CJC infinity_NN1 but_CJC nothing_PNI else_AV0 in_PRP a_AT0 step_NN1 function_NN1 ._SENT In_PRP the_AT0 classical_AJ0 language_NN1 of_PRF membrane_NN1 biophysics_NN2 ,_PUN the_AT0 ionic_AJ0 conductances_NN2 gNa_NP0 and_CJC gK_NP0 must_VM0 not_XX0 depend_VVI on_PRP the_AT0 rate_NN1 of_PRF change_NN1 of_PRF voltage_NN1 ._SENT 
Much_DT0 work_NN1 has_VHZ been_VBN conducted_VVN showing_VVG that_CJT step_NN1 functions_NN2 are_VBB enough_AV0 to_TO0 understand_VVI the_AT0 voltage_NN1 dependent_AJ0 mechanisms_NN2 in_PRP the_AT0 classical_AJ0 action_NN1 potential_NN1 of_PRF the_AT0 squid_NN0 axon_NN1 ,_PUN starting_VVG with_PRP [_PUL 78_CRD ]_PUR ,_PUN Figure_NN1 10_CRD and_CJC Equation_NN1 (_PUL 11_CRD )_PUR ._SENT Hodgkin_NP0 kindly_AV0 explained_VVD the_AT0 significance_NN1 of_PRF this_DT0 issue_NN1 to_PRP colleagues_NN2 ,_PUN including_PRP the_AT0 author_NN1 (_PUL around_PRP 1970_CRD )_PUR ._SENT He_PNP explained_VVD the_AT0 possible_AJ0 incompleteness_NN1 of_PRF step_NN1 function_NN1 measurements_NN2 :_PUN if_CJS sodium_NN1 conductance_NN1 had_VHD a_AT0 significant_AJ0 dependence_NN1 on_PRP ∂V_NP0 /_PUN ∂t_NP0 ,_PUN the_AT0 action_NN1 potential_NN1 computed_VVN from_PRP voltage_NN1 clamp_NN1 data_NN0 would_VM0 differ_VVI from_PRP experimental_AJ0 measurements_NN2 ._SENT He_PNP mentioned_VVD that_CJT this_DT0 possibility_NN1 was_VBD an_AT0 important_AJ0 motivation_NN1 for_PRP Huxley_NP0 's_POS heroic_AJ0 hand_NN1 integration_NN1 of_PRF the_AT0 Hodgkin_NP0 Huxley_NP0 differential_AJ0 equations_NN2 ._SENT Huxley_NP0 confirmed_VVD this_DT0 in_PRP a_AT0 separate_AJ0 personal_AJ0 communication_NN1 ,_PUN Huxley_NP0 to_PRP Eisenberg_NP0 ._SENT Those_DT0 computations_NN2 and_CJC many_DT0 papers_NN2 since_PRP [_PUL 89_CRD ,_PUN 90_CRD ,_PUN 91_CRD ]_PUR have_VHB shown_VVN that_CJT voltage_NN1 clamp_NN1 data_NN0 (_PUL in_PRP response_NN1 to_PRP steps_NN2 )_PUR is_VBZ enough_AV0 to_TO0 predict_VVI the_AT0 shape_NN1 and_CJC propagation_NN1 of_PRF the_AT0 action_NN1 potential_NN1 in_PRP nerve_NN1 and_CJC skeletal_AJ0 muscle_NN1 ._SENT It_PNP should_VM0 be_VBI clearly_AV0 understood_VVN that_CJT such_DT0 a_AT0 result_NN1 is_VBZ not_XX0 available_AJ0 for_PRP biological_AJ0 systems_NN2 in_PRP which_DTQ the_AT0 influx_NN1 of_PRF Ca++_NP0 drives_VVZ the_AT0 action_NN1 potential_NN1 and_CJC its_DPS propagation_NN1 ._SENT 
The_AT0 conductance_NN1 of_PRF the_AT0 voltage_NN1 activated_VVN calcium_NN1 channel_NN1 has_VHZ complex_AJ0 dependence_NN1 on_PRP the_AT0 current_NN1 through_PRP the_AT0 channel_NN1 because_CJS the_AT0 concentration_NN1 of_PRF Ca++_NP0 in_PRP the_AT0 cytoplasm_NN1 is_VBZ so_AV0 low_AJ0 (_PUL ~_NN1 10−8_CRD M_ZZ0 at_PRP rest_NN1 )_PUR that_CJT the_AT0 current_NN1 almost_AV0 always_AV0 changes_VVZ the_AT0 local_AJ0 concentration_NN1 in_PRP and_CJC near_PRP the_AT0 channel_NN1 on_PRP the_AT0 cytoplasmic_AJ0 side_NN1 ._SENT Those_DT0 concentration_NN1 changes_NN2 ,_PUN in_PRP turn_NN1 ,_PUN alter_VVB the_AT0 gating_NN1 and_CJC selectivity_NN1 characteristics_NN2 of_PRF the_AT0 channel_NN1 protein_NN1 ,_PUN as_CJS calcium_NN1 ions_NN2 are_VBB prone_AJ0 to_TO0 do_VDI int_VVB many_DT0 physical_AJ0 and_CJC biological_AJ0 systems_NN2 ,_PUN particularly_AV0 at_PRP interfaces_NN2 ._SENT 
It_PNP seems_VVZ unlikely_AJ0 that_CJT the_AT0 resulting_AJ0 properties_NN2 of_PRF voltage_NN1 dependent_AJ0 calcium_NN1 channels_NN2 can_VM0 be_VBI comfortably_AV0 described_VVN by_PRP the_AT0 same_DT0 formalism_NN1 used_VVN for_PRP voltage-controlled_AJ0 sodium_NN1 and_CJC potassium_NN1 channels_NN2 of_PRF nerve_NN1 and_CJC skeletal_AJ0 muscle_NN1 ._SENT That_DT0 formalism_NN1 uses_VVZ variables_NN2 that_CJT depend_VVB on_PRP membrane_NN1 potential_NN1 and_CJC not_XX0 membrane_NN1 current_AJ0 because_CJS Cole_NP0 and_CJC Hodgkin_NP0 guessed_VVD that_CJT neuronal_AJ0 action_NN1 potentials_NN2 were_VBD essentially_AV0 voltage_NN1 dependent_AJ0 ,_PUN not_XX0 current_AJ0 dependent_NN1 ._SENT They_PNP found_VVD action_NN1 potentials_NN2 in_PRP '_PUQ space_NN1 clamped_AJ0 '_POS axons_NN2 with_PRP wires_NN2 down_PRP their_DPS middle_NN1 that_CJT ensured_VVD spatial_AJ0 uniformity_NN1 of_PRF potential_NN1 ._SENT These_DT0 axons_NN2 had_VHD very_AV0 different_AJ0 patterns_NN2 of_PRF current_AJ0 flow_NN1 from_PRP normal_AJ0 axons_NN2 ,_PUN and_CJC so_AV0 Cole_NP0 and_CJC Hodgkin_NP0 were_VBD confirmed_VVN in_PRP their_DPS view_NN1 that_CJT the_AT0 membrane_NN1 processes_NN2 generating_VVG the_AT0 action_NN1 potential_NN1 were_VBD voltage_NN1 dependent_AJ0 ,_PUN much_AV0 more_AV0 than_CJS current_AJ0 dependent_AJ0 (_PUL personal_AJ0 communication_NN1 Cole_NP0 to_PRP Eisenberg_NP0 1960_CRD ;_PUN Hodgkin_NP0 to_PRP Eisenberg_NP0 1961_CRD ,_PUN et_NP0 al._NP0 )_PUR ._SENT 
Hodgkin_NP0 ,_PUN Huxley_NP0 ,_PUN Katz_NP0 ,_PUN and_CJC Cole_NP0 did_VDD not_XX0 know_VVI of_PRF action_NN1 potentials_NN2 driven_VVN by_PRP calcium_NN1 channels_NN2 ,_PUN nor_CJC of_PRF the_AT0 extraordinarily_AV0 small_AJ0 concentration_NN1 of_PRF calcium_NN1 ions_NN2 inside_PRP cells_NN2 ._SENT There_EX0 may_VM0 of_PRF course_NN1 be_VBI other_AJ0 reasons_NN2 the_AT0 formalism_NN1 is_VBZ inadequate_AJ0 ._SENT In_PRP summary_NN1 ,_PUN experiments_NN2 ,_PUN theory_NN1 ,_PUN computations_NN2 and_CJC perhaps_AV0 simulations_NN2 are_VBB needed_VVN to_TO0 show_VVI that_DT0 responses_NN2 to_PRP steps_NN2 of_PRF voltage_NN1 allow_VVB computation_NN1 of_PRF a_AT0 calcium_NN1 driven_VVN action_NN1 potential_NN1 ._SENT 
The_AT0 polarization_NN1 protocol_NN1 described_VVN here_AV0 can_VM0 be_VBI applied_VVN to_PRP simulations_NN2 of_PRF polarization_NN1 as_CJS well_AV0 as_CJS experimental_AJ0 measurements_NN2 of_PRF polarization_NN1 ._SENT Indeed_AV0 ,_PUN the_AT0 operational_AJ0 definition_NN1 of_PRF polarization_NN1 has_VHZ been_VBN applied_VVN even_AV0 when_CJS theories_NN2 or_CJC simulations_NN2 are_VBB enormously_AV0 complicated_AJ0 by_PRP atomic_AJ0 detail_NN1 that_CJT includes_VVZ the_AT0 individual_AJ0 motions_NN2 of_PRF thousands_CRD of_PRF atoms_NN2 ._SENT 
Another_DT0 question_NN1 of_PRF general_AJ0 interest_NN1 is_VBZ how_AVQ does_VDZ the_AT0 polarization_NN1 defined_VVN this_DT0 way_NN1 correspond_VVB to_PRP the_AT0 polarization_NN1 −P_NN1 =_NN1 (_PUL εr−1_CRD )_PUR ε0E_NN1 in_PRP the_AT0 classical_AJ0 formulation_NN1 of_PRF the_AT0 Maxwell_NP0 Equations_NN2 (_PUL 7_CRD )_PUR and_CJC implied_VVN in_PRP Equations_NN2 (_PUL 1_CRD )_PUR and_CJC (_PUL 6_CRD )_PUR ?_SENT Does_VDZ the_AT0 estimated_AJ0 polarization_NN1 equal_AJ0 P_ZZ0 ?_SENT 
The_AT0 answer_NN1 is_VBZ not_XX0 pleasing_AJ0 ._SENT Polarization_NN1 cannot_NN1 be_VBI defined_VVN in_PRP general_AJ0 ._SENT The_AT0 variety_NN1 of_PRF possible_AJ0 responses_NN2 of_PRF matter_NN1 to_PRP a_AT0 step_NN1 of_PRF potential_NN1 prevents_VVZ a_AT0 general_AJ0 answer_NN1 ._SENT Indeed_AV0 ,_PUN a_AT0 main_AJ0 point_NN1 of_PRF this_DT0 paper_NN1 is_VBZ that_CJT polarization_NN1 must_VM0 be_VBI defined_VVN by_PRP a_AT0 protocol_NN1 in_PRP a_AT0 specific_AJ0 setting_NN1 that_CJT specifies_VVZ how_AVQ the_AT0 local_AJ0 electric_AJ0 field_NN1 changes_VVZ the_AT0 distribution_NN1 of_PRF charge_NN1 ._SENT 
Polarization_NN1 cannot_NN1 be_VBI defined_VVN in_PRP general_AJ0 because_CJS there_EX0 are_VBB too_AV0 many_DT0 possible_AJ0 motions_NN2 of_PRF mass_NN1 with_PRP charge_NN1 in_PRP response_NN1 to_PRP a_AT0 change_NN1 in_PRP the_AT0 electric_AJ0 field_NN1 ._SENT Every_AT0 possible_AJ0 motion_NN1 of_PRF mass_NN1 (_PUL with_PRP charge_NN1 )_PUR ,_PUN including_PRP rotations_NN2 and_CJC translations_NN2 and_CJC changes_NN2 of_PRF shape_NN1 and_CJC density_NN1 of_PRF charge_NN1 ,_PUN would_VM0 produce_VVI a_AT0 polarization_NN1 ._SENT Polarization_NN1 currents_NN2 can_VM0 be_VBI as_AV0 complicated_AJ0 as_CJS the_AT0 motions_NN2 of_PRF matter_NN1 ._SENT 
In_PRP mechanical_AJ0 systems_NN2 in_PRP general_AJ0 these_DT0 issues_NN2 do_VDB not_XX0 attract_VVI much_DT0 attention_NN1 ._SENT It_PNP seems_VVZ obvious_AJ0 that_CJT one_PNI must_VM0 have_VHI a_AT0 model_NN1 and_CJC theory_NN1 of_PRF how_AVQ a_AT0 system_NN1 changes_NN2 shape_NN1 (_PUL and_CJC distribution_NN1 of_PRF mass_NN1 )_PUR when_CJS forces_NN2 are_VBB applied_VVN ._SENT Seeking_VVG a_AT0 general_AJ0 treatment_NN1 is_VBZ silly_AJ0 ._SENT In_PRP electrodynamics_NN2 ,_PUN for_PRP illogical_AJ0 reasons_NN2 of_PRF history_NN1 ,_PUN tradition_NN1 ,_PUN and_CJC respect_NN1 for_PRP our_DPS elders_NN2 ,_PUN scientists_NN2 have_VHB sought_VVN the_AT0 general_AJ0 treatment_NN1 that_CJT would_VM0 be_VBI considered_VVN silly_AJ0 for_PRP mechanical_AJ0 systems_NN2 ._SENT 
