Impact of Adaptation Currents on Synchronization of Coupled Exponential Integrate-and-Fire Neurons
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{"title"=>"Impact of adaptation currents on synchronization of coupled exponential integrate-and-fire neurons", "type"=>"journal", "authors"=>[{"first_name"=>"Josef", "last_name"=>"Ladenbauer", "scopus_author_id"=>"36671571500"}, {"first_name"=>"Moritz", "last_name"=>"Augustin", "scopus_author_id"=>"55220817200"}, {"first_name"=>"Lie June", "last_name"=>"Shiau", "scopus_author_id"=>"7004998869"}, {"first_name"=>"Klaus", "last_name"=>"Obermayer", "scopus_author_id"=>"7003506023"}], "year"=>2012, "source"=>"PLoS Computational Biology", "identifiers"=>{"arxiv"=>"arXiv:1310.2430v1", "scopus"=>"2-s2.0-84861133624", "pui"=>"364830882", "pmid"=>"22511861", "issn"=>"1553734X", "isbn"=>"1553-734X", "doi"=>"10.1371/journal.pcbi.1002478", "sgr"=>"84861133624"}, "id"=>"0750d919-e65c-3148-89a1-d720db9ec50a", "abstract"=>"The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.", "link"=>"http://www.mendeley.com/research/impact-adaptation-currents-synchronization-coupled-exponential-integrateandfire-neurons", "reader_count"=>67, "reader_count_by_academic_status"=>{"Unspecified"=>2, "Professor > Associate Professor"=>4, "Researcher"=>12, "Student > Doctoral Student"=>2, "Student > Ph. D. Student"=>26, "Student > Postgraduate"=>1, "Student > Master"=>5, "Other"=>3, "Student > Bachelor"=>4, "Lecturer"=>1, "Lecturer > Senior Lecturer"=>1, "Professor"=>6}, "reader_count_by_user_role"=>{"Unspecified"=>2, "Professor > Associate Professor"=>4, "Researcher"=>12, "Student > Doctoral Student"=>2, "Student > Ph. D. Student"=>26, "Student > Postgraduate"=>1, "Student > Master"=>5, "Other"=>3, "Student > Bachelor"=>4, "Lecturer"=>1, "Lecturer > Senior Lecturer"=>1, "Professor"=>6}, "reader_count_by_subject_area"=>{"Engineering"=>8, "Unspecified"=>2, "Mathematics"=>5, "Agricultural and Biological Sciences"=>18, "Arts and Humanities"=>1, "Neuroscience"=>12, "Business, Management and Accounting"=>1, "Physics and Astronomy"=>14, "Psychology"=>1, "Computer Science"=>4, "Linguistics"=>1}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>8}, "Neuroscience"=>{"Neuroscience"=>12}, "Physics and Astronomy"=>{"Physics and Astronomy"=>14}, "Psychology"=>{"Psychology"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>18}, "Computer Science"=>{"Computer Science"=>4}, "Business, Management and Accounting"=>{"Business, Management and Accounting"=>1}, "Linguistics"=>{"Linguistics"=>1}, "Mathematics"=>{"Mathematics"=>5}, "Unspecified"=>{"Unspecified"=>2}, "Arts and Humanities"=>{"Arts and Humanities"=>1}}, "reader_count_by_country"=>{"Netherlands"=>1, "Turkey"=>1, "Norway"=>1, "United States"=>1, "Japan"=>1, "United Kingdom"=>1, "France"=>2, "Germany"=>1}, "group_count"=>6}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/654404"], "description"=>"<p>Spike times (solid bars) of two neurons oscillating with a small phase difference and coupled through excitatory (A and B) or inhibitory synapses (C and D) with a symmetric conduction delay . The PRCs of the neurons that make up each pair are displayed below. In A and C the neurons have type I PRCs, in B and D the PRCs are type II. The time (phase) at which each neuron receives a synaptic current is shown along the spike trace. Phase advances or delays, considering the time of input arrival and the shape of the PRC, are indicated by advanced or delayed subsequent spike times. Dashed bars indicate spike times without synaptic inputs. The consequent changes in are highlighted.</p>", "links"=>[], "tags"=>["conduction", "delays", "synchrony", "coupled"], "article_id"=>324896, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g007", "stats"=>{"downloads"=>0, "page_views"=>10, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effects_of_conduction_delays_on_the_stability_of_synchrony_in_coupled_pairs_/324896", "title"=>"Effects of conduction delays on the stability of synchrony in coupled pairs.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:21:36"}
  • {"files"=>["https://ndownloader.figshare.com/files/654069"], "description"=>"<p>A,B: Rheobase current (solid black), SN and AH bifurcation currents , (dashed grey, dashed black) respectively, as well as input current (green) which separates type I (blue) and type II (yellow) PRC regions, as a function of , for (A) and (B). At a BT bifurcation occurs at (where the SN and the AH bifurcations meet) marked by the red dot. The region around is displayed in a zoomed view. If the system undergoes a SN bifurcation at , if an AH bifurcation occurs at . C,D: Spike frequencies corresponding to the input currents in A and B. Note that the region in - space where the PRCs are type II is very shallow in A compared to B, the corresponding regions in - space shown in C and D however are rather similar. This is due to the steep (flat) - relationship for () respectively (see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002478#pcbi-1002478-g001\" target=\"_blank\">Fig. 1D,E</a>). E,F: PRCs with locations in - space as indicated, scaled to the same period .</p>", "links"=>[], "tags"=>["currents", "aeif", "dependence", "prc", "characteristics"], "article_id"=>324560, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g003", "stats"=>{"downloads"=>0, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Bifurcation_currents_of_the_aEIF_model_and_dependence_of_PRC_characteristics_on_the_input_current_/324560", "title"=>"Bifurcation currents of the aEIF model and dependence of PRC characteristics on the input current.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:16:00"}
  • {"files"=>["https://ndownloader.figshare.com/files/654141"], "description"=>"<p>A: PRC of an aEIF neuron (top) spiking at and interaction functions (bottom) obtained for synaptic conductances with three different sets of synaptic time constants: , (blue), , (green); , (magenta), and . The synaptic current associated with each pair of time constants (center) illustrates the three synaptic timescales relative to the period . Note that shown here is received by the neuron at the beginning of its ISI. B: PRC (solid black) of an aEIF neuron spiking at and excitatory synaptic currents with , (dashed blue) received at three different phases. Assuming the input comes from a second, synchronous neuron, these phases represent three different conduction delays , , and . Note that synaptic input received at an earlier phase causes a larger peak of , due to the smaller value of the membrane potential which leads to a larger difference to the synapse's reversal potential . C: Interaction functions for pairs of neurons with the PRC shown in B, coupled by excitatory synapses with , , and delays and . The values of at are highlighted by blue circles. The slopes of , in terms of both left and right sided limits and , indicate whether the synchronous states are stable or unstable (see main text).</p>", "links"=>[], "tags"=>["prc"], "article_id"=>324638, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g004", "stats"=>{"downloads"=>0, "page_views"=>12, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Relationship_between_the_PRC_and_the_interaction_function_/324638", "title"=>"Relationship between the PRC and the interaction function.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:17:18"}
  • {"files"=>["https://ndownloader.figshare.com/files/654499"], "description"=>"<p>A–C: Change of phase difference given by equation (25), as a function of for pairs of excitatory aEIF neurons coupled with different ratios of synaptic conductances (). Zero crossings with a negative slope indicate stable phase locking and are marked by black dots. Adaptation parameters of the neurons and PRCs are shown in the top row. D–I: Stable phase locked states of excitatory (D–F) and inhibitory (G–I) pairs as a function of the synaptic conductance ratio, for three different conduction delays , and (black, brown, green). Unstable states are not shown for improved clarity. Dashed lines denote equal synaptic strengths, grey arrows indicate a continuous increase or decrease of (mod ) for ratios at which phase locked states do not exist (see main text).</p>", "links"=>[], "tags"=>["locking", "aeif", "pairs", "coupled", "delays", "heterogeneous", "synaptic"], "article_id"=>324986, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g008", "stats"=>{"downloads"=>0, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Phase_locking_of_aEIF_pairs_coupled_with_delays_and_heterogeneous_synaptic_strengths_/324986", "title"=>"Phase locking of aEIF pairs coupled with delays and heterogeneous synaptic strengths.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:23:06"}
  • {"files"=>["https://ndownloader.figshare.com/files/335889"], "description"=>"<div><p>The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.</p> </div>", "links"=>[], "tags"=>["adaptation", "currents", "synchronization", "coupled", "exponential", "integrate-and-fire", "neurons"], "article_id"=>126359, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Impact_of_Adaptation_Currents_on_Synchronization_of_Coupled_Exponential_Integrate_and_Fire_Neurons/126359", "title"=>"Impact of Adaptation Currents on Synchronization of Coupled Exponential Integrate-and-Fire Neurons", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2012-04-12 01:45:59"}
  • {"files"=>["https://ndownloader.figshare.com/files/654329"], "description"=>"<p>Stable (solid black) and unstable (dashed grey) phase locked states of aEIF pairs without adaptation, (A and D), and with adaptation, , (B and E), , (C and F), as a function of the conduction delay . The neurons are coupled through excitatory (A–C) or inhibitory synapses (D–F) with equal conductances (). Synaptic time constants are as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002478#pcbi-1002478-g005\" target=\"_blank\">Fig. 5</a>. Circles denote steady-state phase differences of numerically simulated pairs of aEIF neurons. The corresponding PRCs are shown in the top row. was 25 ms.</p>", "links"=>[], "tags"=>["locking", "coupled", "aeif", "pairs", "conduction"], "article_id"=>324816, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g006", "stats"=>{"downloads"=>0, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Phase_locking_of_coupled_aEIF_pairs_with_conduction_delays_/324816", "title"=>"Phase locking of coupled aEIF pairs with conduction delays.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:20:16"}
  • {"files"=>["https://ndownloader.figshare.com/files/654208"], "description"=>"<p>Stable (solid black) and unstable (dashed grey) phase locked states of pairs of aEIF neurons spiking at with identical PRCs as a function of adaptation parameters. These phase locked states were obtained by evaluating the interaction function. Circles denote the steady-state phase differences by numerically simulating pairs of aEIF neurons according to eqs. (1)–(3). To detect bistability, the simulations were run multiple times and the pairs initialized either near in-phase or anti-phase with values of the periodic spiking trajectory. In A and B the neurons are coupled through excitatory, in C and D through inhibitory synapses, as indicated by the diagrams on the left. Synaptic conductances are equal () and conduction delays are not considered here (). Synaptic time constants were , for excitatory and , for inhibitory connections. In A and C, varies from 0 to 0.1 with , whereas in B and D, while varies from 0 to 0.2 nA. All other model parameters are given in the <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002478#s2\" target=\"_blank\">Methods</a> section. The corresponding changes in PRCs are indicated in the top row.</p>", "links"=>[], "tags"=>["adaptation", "locked", "states", "coupled", "aeif"], "article_id"=>324701, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g005", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effects_of_adaptation_on_phase_locked_states_of_coupled_aEIF_pairs_/324701", "title"=>"Effects of adaptation on phase locked states of coupled aEIF pairs.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:18:21"}
  • {"files"=>["https://ndownloader.figshare.com/files/654802"], "description"=>"<p>A–D: Stable (solid black) and unstable (dashed grey) phase locked states of coupled pairs of Traub neurons with identical PRCs, as a function of conductances and , respectively. Corresponding changes in PRCs are displayed in the top row. The neurons are coupled through excitatory or inhibitory synapses as indicated by the diagrams on the left, with equal synaptic strengths, and . E: Network synchronization over time, of coupled excitatory (solid) and inhibitory (dashed) Traub neurons without, (black) or with adaptation, , (blue) and , (red), driven to 40 Hz spiking. The neurons are all-to-all coupled with equal synaptic conductances, (black and blue), (red), but without self-feedback, , and conduction delays, . F: Raster plots showing the spike times during the last 200 ms for the three excitatory networks and the network of inhibitory neurons without adaptation (bottom). The neurons in the columns are sorted according to their last spike time.</p>", "links"=>[], "tags"=>["adaptation", "synchronization", "properties", "traub"], "article_id"=>325291, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g011", "stats"=>{"downloads"=>0, "page_views"=>10, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Influence_of_adaptation_on_synchronization_properties_of_Traub_model_neurons_/325291", "title"=>"Influence of adaptation on synchronization properties of Traub model neurons.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:28:11"}
  • {"files"=>["https://ndownloader.figshare.com/files/654591"], "description"=>"<p>Degree of network synchronization (A) and phase locking (B) of aEIF neurons without adaptation, (black frame) and either adaptation component, respectively, , (blue frame), , (red frame), driven to 40 Hz spiking, all-to-all coupled without self-feedback, for various conduction delays and synaptic conductances. and are random (uniformly distributed) in the indicated intervals. Specifically, , and , , with units in parenthesis. The PRCs of the three neuron types described above are shown in the top row. C: Time course of for networks without delays and equal synaptic strengths, as indicated by the symbols in A. Each and value represents an average over three simulation runs. D: Raster plots for neuron and network parameters as indicated by the symbols in B, where the neurons in the columns are sorted according to their last spike time.</p>", "links"=>[], "tags"=>["adaptation", "aeif"], "article_id"=>325079, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g009", "stats"=>{"downloads"=>0, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Impact_of_adaptation_on_the_behavior_of_aEIF_networks_/325079", "title"=>"Impact of adaptation on the behavior of aEIF networks.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:24:39"}
  • {"files"=>["https://ndownloader.figshare.com/files/654697"], "description"=>"<p>A: Membrane potential of Traub model neurons without adaptation, (black), -mediated, (blue) and -mediated adaptation, (red), in response to step currents , B: the corresponding - curves, and C: the corresponding PRCs. Solid lines in C denote the PRCs, calculated with the adjoint method and scaled by 0.2 mV. Open circles denote the results of numerical simulations of eqs. (4)–(9) with 0.2 mV perturbations at various phases. D,E: Rheobase current (solid black), (dashed grey) and (dashed black), as a function of , for (left) and (right). and converge at marked by the red dot. The input current indicated by the green curve separates type I and type II PRC regions (blue and yellow, respectively). F,G: Spike frequencies according to the input currents in D and E. H,I: PRCs for parametrizations as indicated in F and G (with corresponding to ), scaled to the same period . All other model parameters are provided in the <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002478#s2\" target=\"_blank\">Methods</a> section.</p>", "links"=>[], "tags"=>["adaptation", "spiking", "prcs", "bifurcation", "currents", "traub"], "article_id"=>325189, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g010", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effects_of_adaptation_on_spiking_dynamics_curves_PRCs_and_bifurcation_currents_of_Traub_model_neurons_/325189", "title"=>"Effects of adaptation on spiking dynamics, - curves, PRCs and bifurcation currents of Traub model neurons.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:26:29"}
  • {"files"=>["https://ndownloader.figshare.com/files/653958"], "description"=>"<p>A,B: PRCs associated with adaptation parameters as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002478#pcbi-1002478-g001\" target=\"_blank\">Fig. 1D,E</a>. Solid curves are PRCs calculated with the adjoint method and scaled by 0.1 mV, circles denote PRC points that were obtained from numerical simulations of eqs. (1)–(3), using 0.1 mV perturbations at various phases (see <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002478#s2\" target=\"_blank\">Methods</a> and <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002478#pcbi.1002478.s001\" target=\"_blank\">Text S1</a>). The input currents were chosen to ensure 40 Hz spiking. Note that the discontinuity of the PRCs at is caused by the reset of the spiking trajectories. C–F Top: PRCs for adaptation parameters as indicated and (C), (D), (E), (F). C–F Bottom: Vector field, - and -nullclines, and periodic spiking trajectory in the respective state space. The reset point (solid square) of the trajectory corresponds to the phase . A solid arrow marks the location along the trajectory where the PRC (shown above) has its maximum. Dashed arrows in D, F mark the trajectory points that correspond to the zero crossings of the PRCs. Trajectory points change slowly in regions where the vector field magnitudes are small. The dashed blue curve in D denotes the boundary of the domain of attraction of the fixed point, which is located at the intersection of the nullclines. Note that differences in the vector fields and -nullclines between C and E as well as D and F, are due to the changes in .</p>", "links"=>[], "tags"=>["adaptation", "prcs", "aeif"], "article_id"=>324449, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g002", "stats"=>{"downloads"=>0, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Effects_of_adaptation_on_PRCs_of_aEIF_neurons_/324449", "title"=>"Effects of adaptation on PRCs of aEIF neurons.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:14:09"}
  • {"files"=>["https://ndownloader.figshare.com/files/653892"], "description"=>"<p>A–C: Membrane potential and adaptation current of aEIF neurons without adaptation (A), with subthreshold adaptation (B) and with spike-triggered adaptation (C), in response to step currents . To demonstrate the steep increase of past , was set to . Note that the neuron in C has not reached its steady state frequency by the end of the rectangular current pulse. D,E: - relationships for , (black – blue, D) and , (black – red, E). All other model parameters used for this figure are provided in the <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002478#s2\" target=\"_blank\">Methods</a> section.</p>", "links"=>[], "tags"=>["adaptation", "spiking", "curves", "aeif"], "article_id"=>324383, "categories"=>["Mathematics", "Biological Sciences", "Neuroscience"], "users"=>["Josef Ladenbauer", "Moritz Augustin", "LieJune Shiau", "Klaus Obermayer"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1002478.g001", "stats"=>{"downloads"=>1, "page_views"=>44, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Influence_of_adaptation_on_spiking_behavior_and_curves_of_aEIF_neurons_/324383", "title"=>"Influence of adaptation on spiking behavior and - curves of aEIF neurons.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2012-04-12 01:13:03"}

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